Alpha Strike: Do I have to deal at least one damage to each target at both battlefields, or can I only deal damage to one battlefield?

You can deal damage to units at just one battlefield. You are not required to target units at both battlefields or at every battlefield.

Alpha Strike says "split among enemy units at battlefields" (plural), meaning you can split across multiple battlefields if you want. But because it's a split effect, you choose which enemy units to target, and "any number" includes just one (FAQ #9467, #9488). So you can put all the damage into a single unit at one battlefield if you like.

The only restriction per target: each unit you choose must receive at least 1 damage (Rule 355.14.g, FAQ #11486). You can't assign 0 to a target just to have it "count" — every chosen target gets a positive integer amount.