A ZNY Controller reported a TCAS event that occurred when a VFR aircraft was issued traffic; reported it in sight but continued on a conflicting path that activated an RA.

Narrative

The B737 [was] given a discretionary descent to cross SWEET at FL070 and issued traffic on a BE58 from twenty miles distant on a gradual converging course. The BE58 issued traffic also at twenty miles distant as a B737; converging descending from FL130 to FL070. The BE58 reported traffic in sight. At ten miles; the B737 was reissued the traffic as was the BE58. At three to five miles the BE58 was asked if he had the traffic in sight and did he intend to alter course to miss the traffic. He replied; 'Yes; we're turning'. The B737 continued to descend although it was a discretionary clearance and never requested a vector to avoid VFR traffic. As proximity became very close the B737 climbed to FL085 with an RA and then went down to FL070 when clear of the BE58. 1) The BE58 was responsible to see and avoid. He saw but took no action to avoid. 2) The B737 had the option of stopping his descent above the issued traffic; he opted not to. 3) The B737 had the option of increasing his rate of descent to be below the issued traffic; he opted not to. 4) The B737 had the option of requesting a vector to assist in avoiding as per FARs but made no such request.