This is a fairly common problem on both sides of the equation. We have to continually emphasize the importance of breathing during the techniques with newer students… and the importance of breathing out when having a technique performed on you. So step 1 is to trust that everyone in the room you are practicing has probably had this problem before, either in this or in another art.
As to things that you can do to improve:
When a technique is being performed it is very easy to get "caught up" in either the motion or, more frequently, in some specific part of the motion (e.g., in a strike-then-lock pattern it isn't uncommon to see people get preoccupied with the coming lock and thus throw a halfhearted strike). The goal here is to work to stay present all the way through the technique.
Essentially when doing the technique take it slowly (where you can) and focus on the flow and on how to breathe through the technique. When observing a technique make a note not just of how the bodies move, but also of how the participants breathe.
Take 10 minutes a day and do a breath awareness meditation of some sort. This doesn't have to be super formal (I used to do it on a train), but basically just pay attention to your own breath for ten minutes. Breath by Breath is a good basic guide if you are interested in the Buddhist variation of this practice.
Practice Something that Emphasizes Syncing Breath
Pick up something in addition to Aikido that makes syncing breath and movement explicit, e.g., yoga (or, heck, even running). Essentially find a practice where breathing is not optional and proper breathing is emphasized. Your branch of Aikido or your instructors may even have such a practice built-in which you can do on your own (hapkido has danjeon breathing).