Have you tried a dream journal? We forget most of our dreams because we might have them at 2 am and wake up at 7 am. If you wake yourself up in the middle of the night one or two times, you're more likely to have been in the middle of a dream, and it's still up there in your brain enough to write down. The more you do this, the easier it becomes.
Personally I strongly do not want to get better at remembering dreams. At the moment I very rarely remember anything about dreaming, and on the very rare occasion that some fragment of memory from a dream pops into my head it is super confusing until I identify "oh, that must have been from a dream". I prefer to keep my memory uncontaminated with random garbage :)
I remember my dreams quite well. Years ago, I did a dream journal to up that even further. At the time, I discussed doing so with a friend, and she expressed a similar sentiment to yours. In our discussion, she explained not wanting to "carry emotional baggage" from a dream into her day, being distracted by it, and so forth.
That phrasing of "carrying emotional baggage" stuck with me, because together we realized that people can relate to their dreams very differently. If she remembers a dream, she remembers the feelings and feels them all over again. I regard dreams as junk data, and can't imagine "feeling" anything about one longer than a few moments after I wake.
