On page 14 of FAS.org it indicates his salary was initially set at $10,000. Since that exceeded his previous UC professor’s salary he asked that it be reduced in line with others. Apparently the Regents turned down his request (following letter on that page).
As a comparison, on page 10 there is a salary listing for "Persons not now holding an academic position but who were in academic work", where the maximum salary (presumed monthly) for a PhD with more than 4 years of experience is set at $400, or $4800/year. So, Oppenheimer's salary was roughly double that of an experienced PhD from outside academia.
Now, at the US National Lab where I work, an experienced PhD will be paid somewhere around ~30 times the $4800/yr, or Oppenheimer's salary would be somewhere in the neighborhood of $300k/yr (plus/minus $100k?).
As a separate comparison, per Business Insider, a US Army private was paid $50/month during the war. A technician at Los Alamos made up to six times that salary (and was not shot at).