The downfall of the purse seems concurrent with the rise of the pocket.

The closely fitted clothing prevalent up to the end of the fifteenth century does not lend itself well to the insertion of a pocket, so the pocket lived on the outside of the clothing in the form of a purse or pouch. However, from the sixteenth century onwards clothing became looser and the internal pocket developed. It has been suggested that the internal pocket was developed by sailors, and it is certainly true that the earliest depictions of pockets are often found on illustrations of seamen: see, for example, the engravings of Theodore de Bry. One of the oldest surviving European pockets is on a jerkin recovered from the 1545 wreck of the Mary Rose. Whether the theory is correct or not, I wouldn't like to say, but I see no evidence to contradict it. Logically too, the internal pocket would be much safer and more practical than a flappy external pouch in the close working confines of a ship.

Obviously there was a long period of cross-over when pockets and purses were both in use, but on the whole purses became rarer in the sixteenth century and were almost non-existent by the end of the seventeenth century.

Disclaimer: I'm talking purely about English fashions, I have little knowledge of the case anywhere else.