Army then. Marines are Navy right? Depending on the job you choose, you may be required to swim/dive, which isn't good for the asthma. Army personell however, have a higher standard of fitness, and the annual fitness tests are harder than Navy and Airforce.
Go with College, if you have a passion for a Military try to get into the Naval Academy or West Point, just try to give your lungs some time for the asthma to where off. I also have asthma and my doctor told me to take my meds and give it some time.