I have asthma but it's for the most part asymptomatic. Is it ok for me to take aleeve?

Likely yes. There is a rare form of severe asthma where people are sensitive to medicines like aspirin, ibuprofen, or Aleve (naproxen). These people usually also have very bad sinusitis as well. If you know you have done well with Aspirin or Ibuprofen in the past without problems, you most likely can do well with aleeve.
Yes. If you don't have problems with Aleve (naproxen) it it should be fine. Just take it with food to protect your stomach!
Yes. A small percentage of people have sensitivity to aspirin, which causes a flare- of asthma. For those people, medications like Aleve or Ibuprofen can cause breathing problems. Tylenol (acetaminophen) is generally considerd safe. However, the percentage of people with asthma who also react to anti-inflammatory medications is low, and the use of these medications is usually safe.