For example, if sentient life exists elsewhere in the universe, your odds of being a human are vanishingly small. This would suggest sentient life does not exist elsewhere in the universe.

That's not how the anthropic principle works.

The anthropic principle lets you compute the posterior probability of some value V of the world, given an observable W. The observable can be the number of humans who have lived so far, and the value V can be the number of humans who will ever live. The probability of a V where 100W < V is smaller than the probability of a V only a few times larger than W.

It's unclear if you get to count transhumans and AIs in V, which is the same problem Yvain is raising here about whether to include bats and ants in the distribution.

You can't conclude that there aren't other planets with life because you ended up here, because the probability of different values of V doesn't depend on the observable W. There's no obvious reason why P(there are 9999 other planets with life | I'm on this planet here with life) / P(there are 9999 other planets with life) would be different than P(there are 0 other planets with life | I'm on this planet with life) / P(there are 0 other planets with life).

(I divided by the priors to show that the anthropic principle takes effect only in the conditional probability; having a different prior probability is not an anthropic effect.)

Disclaimer: I'm a little drunk.

I'm troubled now that this formulation doesn't seem to work, because it relies on saying "P(fraction of all humans who have lived so far is < X)". It doesn't work if you replace the "<" with an "=". But the observable has an "=".

BTW, outside transhumanist circles, the anthropic principle is usually used to justify having a universe fine-tuned for life, not to figure out where you stand in time, or whether life will go extinct.