Puncture proof bike tires work extremely well. I have Continental Top Contact tires. Except for overinflation (my fault), never had a flat over thousands of miles. They feel nice to ride on too. Before that, I had Panaracer ribmo tires. No flats, ever. However, the ride was a little bit stiff.

Both were a little heavy, but worth the extra weight.

If you're getting flats, that's a much better solution.

We were given some airless tires a few years ago by two different manufacturers to test on some bicycle trailers. Our experience was the tires performed find when the trailers were empty, but the rolling resistance was noticeably higher than a pneumatic tire when the trailer was loaded--so much more, in fact, that we discarded them immediately after the test.

I think the best solution is to use pneumatic tires with a Kevlar liner built into the tire casting.

Rent-a-bikes in my city actually use airless tires, so they seem to already exist for this purpose. I don't know what is inside, but from the outside they look like normal bike tires. The ride is noticeably less comfortable though.