If they are happening on the bottom then it's probably blossom end rot which can easily be fixed. Adequate and consistent watering is needed to avoid blossom end rot and calcium is also necessary. If you take a couple of eggshells and break them up and sprinkle them around each plant this should help with the calcium issue. As far as watering, just make sure that the tomatoes have consistent watering throughout the week, mulching should help.
If the dark spots are happening towards the top near the stem, then I'm afraid it's Blight, which is a bit harder to control but not impossible. The best thing I could recommend for this is to spray with a copper or sulfur spray to control the fungus. Keeping the leaves dry and with good circulation is key to avoiding this. For next year, make sure you either solarize the soil or rotate your crops.
If they are in random places then it could either a black mold or anthracnose. Both are usually caused by water splashing onto the fruits and not being allowed to dry quickly enough. Both can be controlled with a copper or sulfur spray and can be avoided by not allowing water to splash on the fruits.
Good luck!
ETA: I didn't read when you said happening near the stem...
You could but they may not ripen correctly or you may lose flavor. If you have some that are starting to turn but are not developing blight than you should pick them and ripen them in a dark warm place preferably with other ripened tomatoes. A brown paper bag on a high shelf works well for this.
I would highly recommend getting a copper or sulfur spray asap. For now you can mix milk and water (using fat free milk at a ratio of 1:1 or whole milk at a ration of 5:1) and spray your fruits and leaves with this mixture to help prevent blight.