Interestingly enough, while Baxter and April have a connection in many verses with April serving as his assistant, the 1987 series is one verse where that is not the case. Yet I am particularly fascinated by having them interact anyway.
April first became aware of Baxter's existence in season 1, when the Turtles asked her to find information on him. Before the night was out, her entire apartment building was destroyed by Mousers and she believed that Baxter had deliberately tried to take over the city with full knowledge of Shredder's murderous schemes. In subsequent episodes they met on several occasions, usually where April was being abducted by him. This happened in It Came From Beneath the Sewers and Return of the Fly. In addition, Baxter and his alien computer friend tried to take over Channel 6 in Son of Return of the Fly II and started turning everyone into giant insects in Revenge of the Fly, including April. Naturally, April would not be eager to trust him after all of that!
Of course, April was wrong about Baxter trying to take over the city with Mousers. He never intended for anyone to be harmed by them at all. But Shredder's frame job was very cruelly effective and put Baxter in the insane asylum for some months. By the time Shredder broke him out, he truly had gone insane. April never canonically met him when he was sane.
In my verse, April first encounters the human and sane Baxter in chapter 4 of Exit the Fly. They share a civil conversation in which April tries to be nice despite having reservations about the Turtles and Splinter sheltering him while he recovers from the cross-fusion. By this point she has been told the truth of what happened in A Thing About Rats, and due to her friendship with the Turtles and Splinter, she is willing to give Baxter a chance since they are.
April believes Baxter should get involved in the fight with Shredder, especially after Barney sides with him. She tries to encourage Baxter to that end, but he feels that pushing Barney about leaving Shredder's employ will only be counterproductive and make everything worse. He also longs to have a normal life and not be part of the Turtles' war with Shredder.
April and Baxter subsequently interact later, when the Turtles and Splinter fall under Barney's mind-control device and Baxter is left watching them abandon the Turtle-Comm and the Lair. He talks with April as he follows them and scoffs at her suggestion that talking to them might help them overcome the mind-control. When he loses them and April catches up first, he follows her and the rest to the Technodrome despite his fears of being harmed by Shredder and Krang.
April witnesses Baxter fighting to overcome his terror enough to help shatter the mind-control, and moments later, she is the one to find him lying hurt after being attacked by a vengeful Barney. She believes he is dead due to overhearing Barney lie to Krang about killing him, and she is sickened and horrified. She rejoices with the Turtles and Splinter to find that Baxter truly is alive and not even seriously hurt, and from this point she considers Baxter a friend. He counts her as one of his true friends along with the Turtles and Splinter.
It is April who arranges for Baxter to obtain a job at Channel 6 as a scientific consultant. She even lies to Burne Thompson about whether Burne has met Baxter before, not wanting Burne's prejudice against mutants to get in the way of Baxter being able to have a decent job. She justifies her actions to Baxter by telling him that Burne never did meet him, only "a poor crazy creature" that was half-human, half-fly. Baxter is touched but worried for April's job. Later, after Burne learns the truth and fires both of them before wanting to rehire Baxter, he insists on making sure April is reinstated as well.
April feels very sad once she knows the whole truth of Baxter's past and has wanted ever since to do an editorial telling everyone said truth so they hopefully won't fear Baxter even after it's announced that he was once the fly that terrorized New York. Baxter has refused the idea, not wanting Barney to get in trouble should Shredder and Krang learn that he lied about killing Baxter. Since they are aware that Baxter is alive now and don't suspect that Barney lied, however, Baxter may finally agree to the interview if April brings it up again. Nevertheless, he is leery about being the subject of such an editorial. He wants to be known for his inventions, not his mistakes.
In spite of the friendship between April and Baxter, there are still misunderstandings and bumps along the way. April is upset when Baxter is initially too terrified to help the Turtles out of a jam involving his upgraded spider robot, not realizing that Baxter has developed acute arachnophobia after having been cross-fused with a fly. She says hurtful things to him, but after learning the truth she feels very badly about it. Baxter feels that April was right about him needing to get involved and isn't upset with her misunderstanding.