It depends upon your definition of 'sunbathing'. There's usually quite a bit of sunshine in Barcelona in April and you should be able to 'sit out' on the beach for a while, but it's unlikely that you'll be 'roasting' in the heat.
I went to Barcelona in the second half of April last year. (Just a day trip, from Stansted, on the 20th). I enjoyed a pleasant walk along the sea front and spent half an hour sitting on the beach but there were very few others keeping me company. It was 'warm' but not 'hot'.
The average maximum daily temperature for Barcelona, in April, is 18C.
http://www.bbc.co.uk/weather/world/city_guides /city.shtml?tt=TT004270
That's about the same temperature forecast for London over the Easter weekend. If you're the type of person who will be sunbathing in Hyde Park this weekend, you'll probably be OK on the Barcelona beaches. Otherwise you might find it a bit cool.
Actually, Barcelona isn't really a recommended location for sunlovers in the near future. The BBC 5 day forecast is predicting rain on every day:
http://www.bbc.co.uk/weather/5day.shtml?world= 0042
The 10 day forecast on uk.weather.com suggests a slight improvement by the end of next week (but not really most people's idea of 'sunbathing weather'):
http://uk.weather.com/weather/local/SPXX0015?f rom=popular
Chris