The drop rates do increase with higher levels, yes. Moreover, on stages 16 and above, you can no longer get xp brews or mystery shards as a reward, so you're guaranteed to get gear (or ancient shards, those have a tiny but constant chance to drop at any stage from 1 to 20).
According to a table I've seen (not official, but based on people who did tons of runs to test), the odds that an artifact drop is 6* start at 5 percent at level 13, increasing to almost 18 percent at level 19 and then 27 percent at level 20 where you can't get lower than 5*. That's if you get an artifact at all, which as mentioned isn't guaranteed below stage 16.