If cloud ceiling or visibility falls below certain minima, the ATC tower will initiate "Low Visibility Procedures (LVP)". These are different from airport to airport and depend largely on the available equipment.
Some airports have surface movement radar (SMR), also known as ground radar, which will let air traffic controllers monitor aircraft on the ground on a dedicated radar screen. At airports with SMR, ground movements are virtually unaffected by LVP.
At airports without SMR, controllers will rely on their mental image of the traffic situation to control aircraft on their ground. (Note: controllers do this even if visibility if OK. You could ask a controller to close their eyes at any point during their work, and they would still know what was going on outside the windows). Obviously, the controller will need to update their mental picture as planes move around. With limited visibility and no SMR, this will mainly be done by pilot reports. Basically, the controller will ask the pilot to inform them when they pass or reach certain parts of the aerodrome, such as a certain taxiway or runway.
One additional piece of equipment that is very common to use during LVP is stopbars. Stopbars are a row of red lights that span across a taxiway and can be switched on and off by the controller. During good visibility conditions, controllers will visually scan the runway area to make sure no aircraft inadvertently enters the runway without permission. Since such visual scanning is not available during low visibility, turning on stopbars near runway holding points serve as an extra reminder to pilots that they have not received permission to proceed.
Low visibility procedures can also include special rules or procedures that come into effect if the visibility becomes really, really bad. For example, where I work, if visibility falls below 200 metres, only one aircrat is allowed to move at the entire airport at a time. So if one aircraft is moving around, all other aircraft have to wait at the gate until that one aircraft is in the air. Obviously, this requires quite a bit of planning work from ATC, but safety always comes first.