Coinbase has stated how they will treat forks and airdrops for Bitcoin, Ethereum, Litecoin, and Bitcoin Cash. Essentially, they will support stable quality forks and airdrops that meet their criteria.
This means if a fork or airdrop passes their Digital Asset Framework (a strict criteria) and is low risk (has active development, liquidity, a lack of bugs, etc.), then they will likely work toward supporting it.
It takes a lot of time and effort for Coinbase to add coins or to create a way for users to withdraw a certain coin, so expect only the major forks and airdrops to get attention. Also, expect a serious lag between the fork and being credited with a fork.
In short, look to the process of Bitcoin Cash as a model (where it was awarded only after months of it performing well in the market and having active development.)
Under this criteria, one might expect to see Bitcoin Gold, Bitcoin Diamond, Litecoin Cash, and a few other forks credited to them, but would not expect to be credited for every fork (there were a lot of forks).
That is the gist, for the official words, see Coinbase’s official blog post’s 1. Our process for adding new assets to Coinbase & GDAX (March 2018 update) and 2. How Coinbase approaches forked and airdropped assets.