Csdrhrt

Just shrink the moon by 6%, and there will never be another total solar eclipse, but new moons will still happen as normal. The moon will be too small to fully cover the sun, so there will be partial and annular eclipses, but no total ones. If you still want the occasional total eclipse, make the moon slightly bigger again, so that eclipses are total if and only if it happens to be at the closest point to the Earth in its orbit at the time of the eclipse. We’d still get the same number of eclipses at the same time, but hardly any of them would be total.

Since both the moon and the Earth are in elliptical orbits, the apparent angular sizes of the moon and sun vary. Specifically, the moon varies between 29' 26" and 33' 30" while the sun varies between 31' 36" and 32' 42". You want the moon’s maximum to be just larger than the sun’s minimum, so that there are very occasional total eclipses if we happen to get an eclipse when the moon is at its closest and the sun is at its furthest. So to reduce the moon’s maximum angular size to 31’ 40” you need to make it 5.5% smaller without changing its orbit.

Keep sun and moon sizes, make the orbit of the Earth around the sun more excentric and with a smaller mean distance.

The new moon is unnafected, but full solar eclipses will only happen if the eclipse happens together with the Earth's apoapsis, or close to it. And that will only happen during a few specific days of the year. Any solar eclipse far from the apoapsis will be partial.

That also implies shorter years and a whole different set of conditions on the planet that might not be compatible with life as we know it, though.

Why not make the moon's orbit precess? If the precession chases the sun (from the perspective of earth) while staying a bit off from it, you could keep new moons as common as always and eliminate solar eclipses altogether.

If you made the presession not quite at the same rate as the sun, you could also cause large periods without a single solar eclipse, followed by a period where a solar eclipse happened every month (which, from a story perspective, could be interesting).

Alternatively, if you had the moon precess in the opposite direction of the earth's movement around the sun (and depending on the length of your lunar month vs solar year), it may be possible to have the moon 'miss' a solar eclipse except for once every few hundreds of years.

Resonance. Pick a ratio of the lunar orbital period to the solar orbital period such that the moon normally in the wrong part of it's orbit when it's both directly sunward and in the plane of the ecliptic. The bigger numbers you need to use to express the ratio the longer it will be between eclipses.

Unlike normal planetary resonance orbits no force will maintain this or cause it to come into being but as the moon slowly spirals out there will be a time that it happens naturally. You'll have to adjust the size of the moon so it provides a total eclipse at that distance from the Earth.

At 90 degree inclination, you could still get new moons near the nodes where the inclination of the moon's orbit with respect to the line from the planet to the sun is low.

The presence of nodes in the planet's orbit where the inclined orbital path of the moon intersects a line pointing towards the sun, however, is not itself problematic, because it is not itself sufficient to cause eclipses. And that's a good thing for you, because it's a simple geometric fact that you can't avoid them. In order to get an eclipse, the moon also has to actually pass through that intersection point during the brief period when it exists every half-year. As such, a very simple solution to avoid ever having an eclipse presents itself: just tweak the moon's orbital period so that it forms a simple integer ratio with the year, such that the moon is always in the same phase on the same date every year--and then just declare that the phases are such that you never have things line up for an eclipse.

Instead of one moon which is just big enough and close enough to cause a total eclipse whenever it intersects between our planet and our sun, why not have several moons which are each either smaller or more distant such that no single moon can produce a full eclipse.
New moons would still happen and would actually be much more common than Earth standard, but total eclipses would only occur when all of the moons simultaneously entered the intersection point, each blocking a portion of the available sunshine and collectively blocking it all.

Another way to make total solar eclipses rarer would be to have a double sun. The planet would orbit both stars in a circumbinary orbit. Perhaps each of the stars and the moon have about the same apparent diameter as seen from the planet. So in a typical eclipse the moon might eclipse only one of the stars and leave the other one unblocked or maybe pass between the two stars as seen from the planet and not eclipse either star.

The two stars would orbit around each other several times during each year of the planet, and the moon would orbit the planet several times during each year of the planet, and the three periods might not have a simple relationship.

Also the three planes - the one in which the two stars orbit, the one in which the planet orbits the center of gravity of the stars, and the one in which the moon obits the planet, might not be the same plane but might be tilted in relation to each other.

Thus there would only be a total eclipse of the two stars when one of the stars is in front of the other one, hiding it as seen from the planet, at the exact moment when the moon is passing in front of the nearer star, hiding it as seen from the planet. And it is possible such such a moment might happen only once in ten years, or once in a hundred years, or once in a thousand years, when everything aligns just right.

Have a moon with a high eccentricity orbit. Have the sizes such that only close it it's perigee (point of closest approach to the earth) is it large enough to get a total solar eclipse. At other points in the orbit it is further away and thus appears smaller in the sky than the sun. Also helped by the fact that orbiting objects move faster at the times of closest approach (due to conservation of energy), so the time spent in the eclipse zone is even smaller.

The more enteric the orbit, the faster the moon shrinks in the sky after each perigee, so the less chance of a total eclipse. The flip side is that more of your new moons will occur when the moon is pretty small in the sky.

Make the moon's orbit very eccentric (as well as very inclined). When it is far from the planet (which is most of the time, given Kepler's Laws), it is too small to cause a total eclipse. The two times a year when the moon passes in front of the sun would have to coincide with the fairly short fraction of its orbit when it is close enough to the planet to totally obscure the sun.

Another option I couldn't find yet would be to make the moon revolve slow enough around Earth that it would always be on the far side, away from the sun. It's an extreme version of the syncing mentioned above. And it could be stable, in the same way the moon's rotation is stable locked to Earth. We'd never have a solar eclipse and lots of new moons. It would be even more stable if Earth was tidally locked to the Sun as the Moon is to Earth.

On a side note, if the moon was in the Lagrange point between the Sun and Earth, we'd never see the Sun...