Aircraft carriers using STOBAR (the skyramp) always need to maintain a speed of about 55kmph in order for the aircraft on them to take off. This is done to generate enough windspeed on the deck to assist the aircraft to take off. And STOBAR system already hampers the carrying capacity of aircraft, so for takeoff from a STOBAR carrier at rest, a further reduction in takeoff weight of the aircraft would be required to generate enough lift for the aircraft to take off at such low speeds. This reduction in weight would depend on the aircraft carrier and the aircraft as such and might create scenarios where this is entirely not possible.
For Aircraft Carriers equipped with CATOBAR, the conventional steam piston drive which is used to catapult the aircraft provides approximately 95 MegaJoules of energy to accelerate the aircraft to speeds where it can takeoff safely. Hence the arguments stated about the STOBAR carriers can be sequentially applied to this class as well. But recent Electromagnetic Aircraft Launch System provides around 122 MegaJoules of energy to accelerate the aircraft. Given the extra amount of energy available to accelerate the aircraft at higher speeds, it is quite possible that it is going to be easier for the aircraft to take off from a static aircraft carrier equipped with electromagnetic catapults when compared to the aircraft carriers equipped with convention steam piston catapults. But the exact details for the feasibility of this idea would be a matter of national interest and might not be available online.
Also, one should not ignore the extra amount of stresses that an aircraft would have to face while taking off from a stationary aircraft carrier as it would have to accelerate to a higher speed at the same available length of the runway and hence this might prove detrimental to the lifetime of the aircraft.
Under research technologies such as Ground Carriers, if implemented would also assist further in reducing the weight of the aircraft and hence make it easier for the aircraft to take off from carriers at much lower speeds.