I recently installed a Pioneer stereo and now my car keeps dying out. Could it be the alternator or battery?

you need to get a NEW battery and assure this NEW battery's cable is solidly connected to the chassis ground....all should be skippy...in skippyville then~

I had three volkswagens catch fire on me...one of them was at the stereo on the freeway....had to pull off the freeway to put it out....it was a cheapo Kraco unit...with potentiometers...still took cassettes then~ one of them I loaned to a friend but did not have the "shorty" small size volkswagen battery under the backseat...the insulation caught fire with this group 27 huge battery in there...I recovered my house keys from the wrecking yard...it was "automatic stickshift" for what that's worth~

yours still has the mechanical float gas gauge on the dash? or was that '64?

6 volt electrical? that could be a problem~