I found it! It's "A Touch of Lavender" by <PERSON>, under the name <PERSON>. It was first published in <PERSON> Science Fiction Magazine, November 1989. It's on the Internet Archive.
I'd got a few details wrong in my recollection.
The story was about aliens who came from the sea. The aliens were intelligent and peaceful.
They are from the sea, they are intelligent and (mostly) peaceful.
The aliens loved listening to music, I think it was something spiritual or of special significance for them.
Sort of true. The aliens have a natural ability to play music organically, which on their home planet is something only priests are permitted to do. The aliens that are on Earth are refugees because they break this rule.
Also, the aliens had a psychoactive, drug-like effect on humans who touched them. 322
I remember it mentioned that there were "junkies" who would touch the aliens to get high.
Yep.
The story was told from the perspective of a child, whose mother develops a relationship with one of these aliens & develops an addiction to the alien's touch. Later the alien starts staying over at their house & sleeping in the mother's bed.
Yep, except the alien starts living with the family first, and the mother later develops the addiction & has the alien sleep in her bed.
I think the alien leaves at some point & goes back to the sea.
This is the biggest thing I got wrong. In actuality, the alien gets murdered by his own kind.
The mother falls pregnant, but I think she denies that she had sex with the alien.
Yep.
At the end of the story, the mother gives birth. The child protagonist notes that the baby doesn't cry, but "listens", implying this is related to the aliens' love of music.
Not quite. This actually happens about halfway through the story, and the baby does cry, but makes no sound while doing so.
At the end of the story,
! The child is heavily implied to be a <PERSON> figure for the aliens, as "one who listens so well that she has no need to speak". The government conspires to have the child given over to the aliens, in exchange for their advanced scientific knowledge. 322