1984 Corvette " No starting life".

Chevy Corvette C-4 1984
150,000 miles

I have put in many hours and dollars into the rebuild of this 1984 Corvette. I just picked it up from it's Spring 2006 Tune - Up and fluid changes. The car wouldn't start, so I had to call AAA to get a boost
everything was closed in town by that time and I drove home, hoping to build up a charge. After that I noticed a peculiar sequence. The Corvette will start on it's on around two starts and then will not start and will need a battery charge, jump. Now the corvette doesn't kick-off at all, lights radio and everthing seems to be working, on battery voltage it states it is 11.8 or something like that, but still no turn over at all. I replaced the battery last summer, alternator around two years ago etc. Do you think this is a starter problem or alternator again.