Some USB/Serial dongles don't support hardware handshake lines at all, some do RTS/CTS, some very sophisticated ones even support DSR/DTR. For those that support the hardware lines, you need to have a proper driver that supports hardware handshake as well (The USB stick needs to convert the pin change to a USB handshake packet and send that to the host and vice versa)
So, the first questions is: Does the chipset on the dongle support hardware handshake? The most common chips known to do that are FTDI USB/Serial chips. In case your dongle has one (and not a cheap Chinese rip-off), chances are high it works.
Another common chip range is the Prolific 2303 - Of these, only the SA, HXD and RA variant seems to support hardware handshake. The others don't (Those are mostly intended for printer interfaces only).
That is the first step to success - The next one is if the Dongle maker managed to expose the hardware handshake lines from the chip - Some do, some don't.
And the next step is whether you got a proper USB driver on the host computer that supports Hardware Handshake. Again, some do, some don't.
Best check very carefully the above points in the product description before you buy such a dongle to get HWHS to work
Tobias