The short answer is no. You can´t make them work under normal circumstances. The PS4 is not compatible with the old accessories and add-ons of the PS3. So in theory, all the sticks from that era are useless.
In practice, however, this is not entirely true. Lab Zero Games developed a driver that allows their games to play with fighting sticks from the last generation. They call it the “Legacy Driver”. In their own words: “If your stick is wired and works without needing special driver software on PC, it should work with Lab Zero’s PS3 stick technology.“
The catch is that it only works on the games that have the driver installed. Luckily, there are a lot of titles with the software. At the very least, the following titles will detect your old stick:
Street Fighter 5
Ultra Street Fighter 4
Mortal Kombat X
Mortal Kombat 11
Guilty Gear Xrd REVELATOR
King of Fighters 14
As you can see, it is quite the list. But, if the game you want to play is not mentioned above, you can´t use your PlayStation 3 fighting stick on it.
You buy an adapter. This one specifically: The Brooks PS3 to PS4 adapter.
Brooks’ adapters let you connect legacy controllers from the PlayStation 3 era to your PlayStation 4. With this gadget, not only you can use your old stick on any game, but also PS3 controllers and steering wheels as well.
The advantage of this adapter is that it will work with all the games on the console´s library, at the price of forking out some thirty to forty dollars to make your controllers work again (which is still cheaper than the price of a high-end arcade stick.)
The last alternative is to buy an arcade stick that will work with both consoles, like this one:
the Hori Fighting Stick Mini. Since it is made to work with both consoles, it will connect without much hassle.
And there you have it. While, in theory, the PS4 is not compatible with the arcade sticks of its predecessor, in practice most major releases are, and you can buy an adapter for the few that aren´t.
Of course, if your stick is made to work on both consoles anyway, this shouldn’t be a problem. You can browse our page here to find more multi-console arcade sticks.