Compare two samples with many zeros

Just an idea (maybe not so good): since about 70% of your values are zeroes, you are actually interested in comparing the last few deciles between groups. One possible tool for that can be the shift function: https://github.com/GRousselet/rogme

But you should indeed clarify your research question, as others said.

Questions: 1. Is your outcome/dependent variable a count data? If so, then you need to see if the zero inflation is actually an inflation in comparison to the conditional mean. If there isn't zero-inflation, then you can try poisson or negative binomial model. If there is zero-inflation, you can try zero-inflated or hurdle poisson (negative binomial) models. 2. What is your exact research question? 3. What's your sample size?

Try plotting a conditional barchart to get a better picture of your data.

I think the supremum should be quite different even when we include 0s and hence K-S stat should be significant. Can you please elaborate, what you mean when you say $p$ values are unstable? Do you mean, they differ vastly across the 10 experiments? Can you share empirical cdf plots?

Also, have you tried ANOVA? It should be able to show that these distributions have different means.

With a lot of zeros in both series, it may be difficult to reject that the null that the means are the same. But you could test for differences in the deciles (or other quantiles) from the two distributions. If the tails are different then the samples are different. For a test see Li, Tiwari and Wells (1996).