Will I be able to go under general anesthesia for a BBL if I have asthma?

If you have mild asthma, you will likely be instructed to use your "as-needed" inhaler on the morning of surgery. You may also be asked to bring this inhaler to the hospital or surgery center with you. If your asthma is more severe, you will be instructed to use your inhalers as usual. If you take any oral medications, you will most likely take those on schedule, too. If you take steroids on a chronic basis (orally) for asthma, you may need more workup and optimization before elective surgery.
Your anesthesiologist will listen to your lungs before you go to sleep to identify any wheezing or other abnormal breath sounds. Your oxygen level will be measured with a lighted finger clip and he or she will review any test results.
During your anesthetic, your anesthesiologist monitors your oxygen level. The little lighted clip placed on your finger is able to measure the percent of oxygen saturation of your red blood cells in your arterial pulse. 
Also, your breathing is monitored while you are anesthetized. During surgery, a ventilator may help you breathe. At the end of surgery, before waking up, you will resume breathing on your own. While the ventilator is being used, the anesthesiologist can get a measurement of your airway pressures. This value will be increased above normal values if your airways are constricted. There are other parameters that can give the anesthesiologist important information about your lungs and breathing, as well.
If your Asthma is as controlled as you said, you should be fine to go forward with your BBL. Please consult your surgeon if you have any more questions. He/She would love to assist you in every aspect of this process.
Thank you for having the courage to ask your question on RealSelf. 
I hope that my answer helps in your plastic surgery journey.  Best Regards, Dr. Naman

   If the asthma is well-controlled, this will be the most important factor in proceeding with surgery.

Any past medical history should be discussed with your surgeon prior to your surgery in full detail for the doctor to best understand your specific medical needs, but if you have asthma that is well controlled then general anesthesia should not pose a problem. Be sure to meet with a board certified plastic surgeon to ensure you are getting the best care possible. Best of luck!