Well, you have to decide if the curse prevents them form continuing actions already in practice, or only affects 'starting' new activities. If they can continue activities they already started. The primary impact would be they might be slow to react to a new situation coming about. This seems to be a judgement call on your part.
If you rule that they basically lose half their rounds worth of actions, you could likely cut the productivity of their downtime actions by about half, and potentially give them a penalty to their practice a trade roll if what they do is highly interrupt driven, such as dealing directly with customers, or dealing with things that are highly time sensitive.
You specifically are saying the 6 seconds piece, leads me to believe you are indicating you are keeping the context of the effect, is constrained to the combat turns scale. But even that doesn't clearly define what should happen if someone starts a multi-turn activity.
Does your activity continue as long as it isn't dependent on reactions to external stimuli. Or do you only make progress on rounds that you succeed the check, considering failed rounds as not progressing, but not terminating the activity. Or does a failed round check on an activity cause the activity to fail, and have to be restarted on a round that they can act?
Alternately, as a turn, is a span of time the player gets to make choices for actions, the amount of time each roll covers might be different for your different modes of play. One interpretation, where Downtime is generally per day, would be that you might make a 50% roll to act on a downtime day. On success you succeed in your downtime, on a failure, you accomplish nothing particular on that turn/day.
Where did you get that curse? I would expect the source might need to have more clarification. [how long is it supposed to impact them, for instance]
As a GM, I try not to curse at my players, but I will curse their characters.
When I do curse their characters, I try to make sure that either someone in the party has the ability to break the curse or provide access to means to break the curse, such as a high level spell caster who wants a task done.
This doesn't really answer your question on "What happens to them while not in combat?" Personally my answer is, life is really inconvenient until they remove the curse, and in game taking steps to remove the curse is the next thing that happens.
Mostly because it's no fun for the player if their character is crippled with the inability to act, and as a GM my role is create a story for the players enjoyment.