What is the current thought about scuba diving for patients with non-exercise induced asthma?

Avoid if pregnant. Even if they don't have non-exercise induced asthma, pregnant women should avoid scuba diving.
For a fit between . Asthma ; diving - the asthma should be mild and well controlled without on-going need for rescue inhaler. Would encourage you to have a diving/hyperbaric physician, pulmonologist or other physician experienced with diving screen you thoroughly. Please note that there may be other physicians who are less rigid about this than I am - but i tend to be risk aversive when it comes to diving.
Not a good idea. The dive alert network (dan) is one of the leading sources of current, reliable information on medical issues related to scuba diving. Here is a link to their article on diving and asthma. http://www.diversalertnetwork.org/medical/articles/article.asp?articleid=22.
It may be OK. The issue with diving & asthma is how well the asthma is controlled. If your symptoms are rare and you are well controlled with only 1 maintenance medication it may be a low risk for you participate in recreational diving. A pulmonary function test gives an idea of how severe your asthma is and how likely you are to have problems while diving. Persons with asthma cannot work as commercial divers.
Criteria vary. In general, asthmatics whose symptoms are triggered by exercise, cold temperature, or stress should not dive. Neither should a person whose asthma is not well-controlled, or who doesn't do well on regular spirometry or post-exercise spirometry. One who needed or used "bronchodilator" medication, or who had asthma symptoms, in the past 48 hrs probably should not dive. One should be physically fit.