Amazing, the OP asks a question relating to 1983. One answer mentions that it could be ex Dan Air crews after the BA takeover. BA took over Dan Air on 8 November 1992. Another states lots of 737 schedules out of Gatwick after the BCAL takeover. That takeover took place in December 1987, BCAL ceased as a legal entity on 14 April 1988. Now someone else is talking about the 1990s.
As we used to be told at school when doing exams over 50 years ago "before answering READ AND UNDERSTAND THE QUESTION"
I added info to chevron's post and, having looked at my log books from 1982 and 1983 can confirm that on the six visits I made to Gatwick during those two years, British Airways, as opposed to British Airtours, 737s were present, between two and four per visit.
I can't comment on how the aircraft and crews were organised. It may be that both Airtours and Airways had common crews at the time. Certainly at that period the vestiges of the 1970s policy of relieving pressure on Heathrow by forcing foreign carriers to use Gatwick were still in being. In an effort to mollify Iberia and the Spanish Government, some BA services to Spain were also moved and it would be these flights the 737s would be used for.