Ceiling Fan Install - detecting voltage in fixture
Non-contact voltage detectors can commonly give false-positive voltage indications from what is known as "phantom voltage".  This alerts you that there may be voltage present in the circuit, but isn't a reliable way to tell which wire the voltage is on or if it's even "real" voltage.

Since the red wire is capped, you should be fine, but if you want to double check with a multi-meter, connect the black lead to a bare ground wire and then touch some non-panted areas of the mounting bracket and fan casing (screw holes are good non-painted areas to test) with the red lead.  Try not to touch any bare metal with your hands while the breaker is on since you are concerned with stray voltage.  You may also want to check the voltage on the red wire to make sure it is independent of the black.  If you get readings of considerably less than 120v, this can also be phantom or induced voltage (like 50v or 60v, etc).