Can I donate blood if I have anemia thalassemia beta trait?

If anemic, no. The red cross won't let people who are anemic donate blood. People are checked for anemia at their screening for blood donation, and if your hemoglobin is less than 12.5 they will not allow you to donate blood. Not everyone with beta thalassemia trait is anemic, so there is a chance that you may pass the screening. However, if you know that you often have a low hemoglobin, expect to be turned away.
Maybe. A person with beta thalassemia trait may be not anemic or mildly anemic. If he is not anemic, he can be a blood donor. However, since he has a more difficult time making hemoglobin (to make new red blood cells), he may not wish to be a blood donor.