Every storm is different, but yes depending on the storm a Tropical Depression 250 miles away can cancel flights or close airports.
Cape Verde is especially vulnerable because of the naturally strong & persistent winds there, even when there are no tropical cyclones nearby.
Remember too that a Tropical Depression is already organized. Even before a Tropical Depression begins, there may already be a tropical disturbance causing thunderstorms in an area hundreds of miles across. Even at this unorganized stage the thunderstorms may cause heavy flooding, spawn tornadoes, etc.
So just because a tropical disturbance doesn't look "close" on the map, it doesn't mean one can be complacent.
Once the disturbance organizes into a tropical cyclone, the affected area could be massive. Sandy was 1,800 km across. Tip generated gale-force winds for over 2,200 km!
Back to Cape Verde, they've already had to permanently close a couple of their island airports due to strong winds. E.g. on Brava they built a new airport in 1992 but had to abandon it just 12 years later in 2004 due to persistent strong winds. They also had to close an airport on Fogo due to the winds there.
The winds there are great for generating electricity and for wind-surfing; but not so ideal for flying.