As the others have pointed out there are many aspects to consider in heat exchanger control, but it seems to me that in this case there are two unique or important factors that require consideration. These are the fact that you are using seawater, and that the exchanger is of the plate type.
I would like to first comment on some of the points already raised. It is almost certain that with careful piping design you can ensure that the water side remains full. As long as your pipers are aware of the requirement it can be addressed.
Recirculation has been raised as a possibility, but you have to consider what you will recirculate. If you recirculate the seawater you will still have a control valve on the hot seawater, plus you will have a large pump probably also made of exotic (expensive) material. ItDepends commented that recirculation of the seawater will give less scaling. The higher velocities will help, but you need to check the temperature effects. He (i.e. ItDepends) also said that recirculation would give lower wall temperatures, but I believe that it would be the opposite. The way recirculation reduces heat transfer is to raise the water temperature, so I think the wall temperatures would actually increase.
Not knowing your process, I may be on the wrong track, but you could also consider recirculating the process side and not controlling the seawater at all. This would result in constant high flowrate and minimum temperature rise on the water side.
RMW said that the reduced flow would reduce the heat transfer. This is the way control valves work and is the desired behavior.
An issue that nobody has raised is the question of equalization of pressures across the plates. I know that modern plate exchangers are rated to hold the design pressures for each pass individually, but I still feel happier without widely different or varying pressures across plates. It's hard to teach an old dog new tricks. Putting the CV on the inlet side will result in wider pressure fluctuations than if it were on the outlet.
As I stated in the beginning of this post, the two important issues are the seawater and the plate exchanger. Again with the caveat of not knowing your process, I would address these issues by not controlling the seawater side (i.e. constant high flow and minimum temp rise) and I would try to get a bypass or recirculation system on the process side that kept the pressure on the process side as constant as possible.