FLT CREW OF AT72 IS SLOW TO REACT TO TA AND A TCASII RA TO WHICH THEY DO NOT RESPOND IN A TIMELY FASHION.

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ATC INFORMED BOTH ACFT OF APCHING TFC AND WE BEGAN A VISUAL SEARCH FOR THE OTHER ACFT. WE WERE ON AN EASTERLY HDG WITH THE AUTOPLT TRACKING DIRECTLY TO THE SJU VOR IN NAV MODE; AND THE OTHER TFC WAS WBOUND. WHEN ATC ISSUED A 010 DEG HDG TO US TO DECONFLICT THE SIT; THE FO (PF) SPUN THE HDG KNOB INTO A L TURN WHILE STILL LOOKING FOR TFC. HE DID NOT SELECT HDG MODE SO THE ACFT DID NOT TURN. I CALLED HIS ATTN TO THE PROB AND IT WAS RESOLVED; BUT THE DELAY IN TURNING ALLOWED THE SEPARATION TO BE REDUCED. I COULD NO LONGER SEE IN THE DIRECTION OF THE TFC DUE TO THE BANK ANGLE. WE WERE IN A L TURN AND THE TFC WAS AT OUR 1-2 O'CLOCK POS. I THEN NOTICED THAT THE FO HAD ONLY MOVED THE HDG BUG TO ABOUT A 030 DEG HDG AND ACTED TO TURN IT THE REST OF THE WAY TO OUR ASSIGNED HDG. THE FO WAS COMPLETELY INVOLVED WITH LOOKING OUT THE WINDOW; AND WHEN THE TCASII ISSUED A DSNDING RA HE TOOK NO ACTION AND SAID NOTHING. I TOOK CTL OF THE ACFT AND ATTEMPTED TO COMPLY WITH THE RA WHICH WAS COMMANDING APPROX 2000 FPM RATE OF DSCNT. I FELT AT THE TIME THAT I DID NOT DUMP THE NOSE AGGRESSIVELY ENOUGH SINCE THE RA CLRED BEFORE I ESTABLISHED THE COMMANDED RATE. ON THE OTHER HAND; ONE OF THE FLT ATTENDANTS LATER SAID THAT SHE HAD NEARLY BEEN LIFTED OFF THE DECK. SHE MIGHT HAVE BEEN INJURED IF I HAD REALLY STUFFED THE NOSE DOWN. THE FO LATER SAID THAT HE HAD THE TFC IN SIGHT WHEN THE RA WAS ISSUED.