This is similar in principle to conventional compressed air energy storage: air is pumped into a reservoir, and the energy later retrieved by expanding the air from the reservoir through turbines (or other air motors) driving generators. The turbine/generator sets may well be the same machines as the pumps.
If the reservoir is placed at the bottom of the sea or a lake, the pressure of water on the outside of the vessel reduces the stress on the walls. If the reservoir is flexible, it can simply collapse as the air is withdrawn, and reinflate as it’s refilled, with the air remaining at almost constant pressure. This avoids the need for strength to keep the water out when the vessel is empty. Because the external pressure on the bottom of the reservoir is higher than that on the top (the difference is the buoyancy of the air) there are some stresses on the container, but they’re very much smaller than the stresses on a normal pressure vessel.
The energy stored in an underwater constant pressure reservoir is more than is stored in a normal, constant volume reservoir. (The reason for this is covered in Underwater Compressed Air Maths.) As well as storing more energy than a rigid air reservoir, this kind of reservoir means you can use a pump/turbine system designed to work with a constant pressure rather than a variable one. Such a system is considerably simpler and more efficient.
Development work on such systems is in progress:
Even more simply, the reservoir could be rigid, with water entering the vessel as the air is consumed, and the water being pushed out by the air as the vessel is filled with air. This scheme would also operate at near constant pressure. In this case there is potential for some loss due to air dissolving in the water, but this would only be an issue for long-term energy storage, and there are ways to reduce the effect almost to zero. There are also potential issues with fouling of the vessel with water plants or animals such as mussels, but again there are ways to avoid this – without resorting to the kind of environmentally unfriendly antifouling used on boats! See Underwater Compressed Air Engineering.
The area of seabed suitable for such systems is vastly greater than the area of land available for pumped storage reservoirs, and most of it is at much greater depths than the elevation of those reservoirs. If such systems can be built economically, they are capable of providing all the energy storage we will need for the foreseeable future, while affecting only a minuscule fraction of the seabed environment, and in a fairly benign way.
Unfortunately for the UK, you have to go a considerable distance offshore to find much really deep water (many other countries have plenty, close inshore), but there’s a fair amount of reasonably deep water in the North Channel (between Scotland and Northern Ireland) and the Firth of Clyde. There are also several lakes, mostly in Scotland, with enough fairly deep water to accommodate a considerable storage capacity – substantially more than the amount of conventional pumped hydro that could reasonably be developed – see Potential Sites.
(Note that such schemes are not generally in any conflict with pumped hydro schemes, even where the same lakes are involved – and in some circumstances may be synergistic with them.)
As with all engineering projects, there are potential hazards; but they are minuscule compared to those associated with fossil fuel or nuclear systems.
 But note, this interesting page refers to a different system, simply pumping the water out of concrete spheres rather than displacing it with compressed air. This requires very thick concrete to resist the pressure of the water, and stores very much less energy – just 20MWh in a 30m sphere at 700m depth instead of the 144MWh that could be stored in compressed air in a sphere with much thinner walls. Their system might perhaps achieve a higher round-trip efficiency, which is important for short-term storage that’s cycled frequently; the compressed air system would probably be a good option for less frequently cycled bulk storage.
 Norway has the Norwegian Trench – thousands of square kilometres, up to 700m deep – off its south coast. It’s an important fishing ground, but only a minuscule fraction of it would be needed for vast energy storage. Spain has enormous areas of water over 4,000m deep not far off its northern coast; France has a substantial area of water over 2,000m deep off its south east. Many but by no means all other countries are well endowed in this respect.
 Both these sites have a potential issue with old munitions which were dumped there – but this is a problem which has already been successfully handled during the laying of submarine communication cables.