This is going to be another long post over time, but I am just starting the topic today and will add to it as I discover more information.
Batman had a long history with aliens, which did not start with Jack Schiff's editorship of the Bat-titles, but certainly did end with him. I will backtrack from the end of Schiff's reign in April 1964; this initial post will cover the alien stories from 1963 and 1964. Note: I will not consider unintelligent alien creatures, largely because that would get too detailed on stories where Batman and Robin visit other worlds.
In Batman #153, Batman, Robin, Batwoman and Batgirl all became the Prisoners of Three Worlds. This was the first book-length story ever in Batman comics, and is a terrific read featuring long segments of the developing romance between Robin and Batgirl (Betty Kane). Here's the key sequence with the alien:
Status: True alien. Bald, green and human-sized with big ears. Score it three Klaatus out of five. Although the story is terrific the alien doesn't add much to it.
This issue does feature a bonus alien race:
Status: True alien. Winged, multi-colored (green, orange and white), bird-like. Four Klaatus out of five.
In Batman #156 (June 1963), Batman and Robin encounter a giant, living, stone idol:
Status: Not an alien; this is a dream sequence.
In Detective #320 (Oct 1963), Bruce and Dick encounter an alien space ship, piloted by a robot:
But this issue turns out to be more of a weird transformation story, as a ray from the space ship turns them green (temporarily). And we never see any actual aliens, so this one doesn't count as a real alien encounter.
In Batman #160 (December, 1963), Batman has a shocking encounter of the third kind:
Status: Not a true alien; this was a crook pretending to be an alien.
Detective #326 features the last of the alien stories, in Captives of the Alien Zoo.
Batman and Robin are captured by aliens who not only run a zoo, but are crooks on the side. Status: True aliens, undersized and yellow-green in color. Rating: As the final alien encounter for Batman, I have to give it five Klaatus out of five. The story is a good one, about how to prove to the aliens that they are intelligent beings and not just dumb animals suitable for circus work.
Julius Schwartz arrived in May 1964, and while he didn't eliminate the science fiction elements from Batman entirely, he did eliminate the cheesy sci-fi; the aliens were gone; except of course for World's Finest and the Justice League. But aside from those stories, I don't think Batman ever encountered another alien.
Update: I should mention here that the style of this post is based loosely on the Comic Treadmill's terrific Prop Stars posts, which explore the various giant props used in Batman stories.