"Eat" is to "feed" as "drink" is to what?
Solution 1:
I don't think there is a single transitive verb for "give drink to [someone]".
If that someone is an animal, you could use water, as in to feed and water a horse:
I didn't go anywhere the next day except up to Grandpa's to feed and water the horse and mule and Granny's chickens.
However, it's unlikely you would use this for a person; the phrase fed and watered refers almost exclusively to livestock, and watered, alone, more to plants, as in I watered the geranium today. It can be used humorously, for example, if you say to a host upon your arrival, to mean that you've already eaten, "No worries; we've already been fed and watered."
If that someone is a baby, you can also suckle them—but this word has even narrower application.
You might try a synonym of sorts for "give," and let the context fill in that a drink was given: (as waitstaff) I served her, I delivered his order; (at the table) I poured her some, I filled his glass.
Solution 2:
Approaching this question etymologically:
"Feed" is not related to "eat" because the former originally means to "foster", "nourish" or "protect". See Etymonline's entry for feed.
Similarly "nourish" has cognates in French "nourrir" (to feed) and "nourriture" (food), but notice how "nourrice" (nanny) refers to the woman who looks after (and used to give milk to) a child. In fact, it is said one "feeds milk to a baby" or "breast-feed", hence it is applicable to liquids too.
It is possible the word "food" narrowed down to non-liquid edibles in time, but in the field of nutrition, "food" refers to drinks as well. See: http://en.wikipedia.org/wiki/Food
Funnily enough, the PIE root of "feed" is *PA- and is also found in Sanskrit as two synonyms √pā which can either mean "to protect" (3rd person indicative present pāti) or "to drink" (3rd person indicative present pibati). See: http://www.sanskrit-lexicon.uni-koeln.de/cgi-bin/monier/serveimg.pl?file=/scans/MWScan/MWScanjpg/mw0612-pazubandhaka.jpg
Etymologically speaking again, "to drench" relates more to "to drink" see: http://www.etymonline.com/index.php?term=drench
So in fact it is not so much "to drink" as it is "to eat" that is in need of a causative. :)