Wet places stay wet
Rainfall happens in an area because of prevailing atmospheric conditions. For example, around the equator there is the Intertropical Convergence zone (ITCZ), a zone of rising humid air that releases rain. This band is around 10 degrees of latitude wide and moves back and forth across the equator. Directly on the equator, there is heavy rainfall all year round, as the ITCZ sits on top of this area for most of the year.
For the tropical rainforests, this is the proximate cause of their enormous moisture. Rising, cooling air releases moisture over the forests. The forests themselves, that great mass of trees and vegetation, then respires the water at night, 'recycling' the water back into the air only for it to fall again. For the Amazon, Congo, and Southeast Asia this is what causes rainfall and is independent of distance from the coast.
Dry places stay dry
On the other hand, around 20-30 degrees north and south, there is a band of descending dry high altitude air. This air is has little to no moisture and thus the land below it is dry year-round. At these latitudes around the world there are the deserts of Southwest US and northern Mexico, the Atamaca and Cuyo in South America, the Sahara and Kalahari in Africa, the Middle East and Thar in Asia, and nearly the whole of Australia.
In these regions, the land is always dry...even if there is a sea breeze! This is especially true in places with cold ocean currents. A desert with a cold ocean current is dry indeed, as you can see from the exceedingly dry conditions in Lima, Peru (Peru current) or Namibe, Angola (Benguela current).
If there is a warmer current, then there is hope for wetter conditions; but only erratically. The best you can hope for is a seasonal burst of rain which leads to a semi-arid climate. Examples here would include scattered summer rains in Mogadishu, Somalia; or scattered winter rains in Benghazi, Libya.
Rain doesn't make it very far inland if it isn't supposed to be wet
The best example for how far rains can penetrate inland would be from Egypt. Here is a map of wind patterns in the Mediterranean.
Egypt sees oncoming winds from the Mediterranean all year round (the ancient Egyptians used this since they could sail up the Nile, but drift with the current back down).
Here (from en.climate-data.org) is a climate breakdown for Alexandria, directly on the coast.
Now here is one for Cairo, about 175 km inland.
Where did all the rain go? As little rain as there was in Alexandria, which is directly on the coast of a warm sea, with oncoming sea breeze all year long, it is all gone by the time the winds get to Cairo. You can tell that Alexandria is getting the sea breeze from a warm sea, because it is slightly warmer than Cairo in the winter, but much cooler in the summer. But wonderful climate aside, its still not getting any rain.
Wet and dry bands are determined by rising and falling air patterns, and by mountains. Places far from the ocean or sea breezes like the middle of the Amazon Basin, or St. Louis, or the center of the Congo, or Chonquing can get plenty of rain if they are located at the right latitudes or on the right side of a continent (the east side is generally wetter than the west; Africa is the big exception). The dryest places in the world are right on the coast in the Atamaca and Namibe deserts.