Relativity
According to relativity, a strong wind with negligible current is locally identical to a negligible wind with a strong current. All of the sailboats that exist today would work just fine in this new environment without any modifications. The issue is they would sail relative to the current.
This means if the current is 30 knots, on the sailboat it would feel like you have 30 knot winds and you could sail using the exact same techniques. The difference would just be that in addition to whatever normal sailing speed you were managing, there's a 30 knot offset in the direction of the current.
With the current
This one is pretty easy. When you're trying to sail with the current, you're effectively trying to sail upwind. There are a lot of boats that can sail upwind at about 60% of the wind speed.
(source)
In this image we can see how fast you can travel on still water with a given wind speed and direction. Basically, you can travel fastest to the sides, at nearly 100% of the wind speed (high performance boats can exceed 100% to the side), around 55% upwind, and 65% downwind.
Then you get to add the current speed back in so you should be able to sail a normal sailboat at about 155% of the speed of the current when you're sailing with the current.
Against the current
If we repeat the math from above now you get that we'll sail at 65% speed downwind (upcurrent) but then when we add the current in now we're going at -35% speed. Uhh Oh.. now we'd be making negative progress.
Solutions
Just deal
If the currents change direction often enough we could just deal with it. 234
If half the time we're making 155% progress and half the time we're making -35% progress we'd still average 120% progress. So we'd still be able to make it to our destination faster than the average current speed.
Propellers
So we'd like to be able to sail downwind faster than the wind. Sounds impossible, but we've actually already achieved something similar: A wind power car than can go 2.86 times the speed of the wind. In the case of the car, the wind pushing on the propeller gives the car thrust to move forward. The wheels then take a portion of this forward energy and direct it back to the propeller to blow backwards. This additional energy allows the wind to provide additional thrust, which then goes into an amplifying cycle only limited by friction and wind resistance (as the vehicle is going faster than the wind so it net blowing against it)
So we can adapt this same technique to a boat, but instead of wheels we'd need a water propeller. The idea would be that the wind would push on the air propeller driving the boat forward. Some of this forward energy would be absorbed by the water propeller which would drive the air propeller into action giving us more thrust to move forward more.
I doubt we'd be able to achieve the amazing 2.86 times the wind speed, but we may still be able to achieve something meaningful using oldmill technology. If this technology worked well, it could actually be used to sail upwind faster as well, by reversing the gear ratios to let the wind propeller power the water propeller.
High Performance Sailing
Turns out it rather than using a typical earth sailboat, you use something like an 18 foot racing skiff, you can sail faster than the wind in pretty much any direction:
With a boat like this you'd have no problem sailing upcurrent. I took the liberty of making a modified graph of how these boats would behave relative to solid ground:
Of course this probably requires modern materials to achieve. 308