I just bought a house in Maryland. It was built int he 50's. Part of the basement has asbestos tile. The previous homeowner removed some of it. But I still have it in my furance room and a room under my front porch. One area the tiles have been patched with cement. And another area there are tiles that are cracked. I don't know if I should remove the tile or put some kind of top coat on them and them put down vinyl on top.

There is probably no harm in leaving it in place, but if it bothers you, I would certainly remove it. Floor tile can be removed using wet methods, and really does not pose much asbestos exposure risk. You can further reduce risks by using wet removal methods and even using ventilation and wearing a cartridge respirator. Obviously there are contractors who can do this job for you. IMO adding a top coat just makes future removal more difficult. Up to you really.

I copied some guidelines from http://www.asbestos-institute.ca/buildings/tiles.html#1 that seem pretty good:
1. Heavy duty wall scraper with approximately 4" blade and 6" and 8" handle.
2. Hammer
3. Commercial type hand held not air blower.
4. Heavy duty impermeable trash bags (or closed impermeable containers), ties and labels.
5. Those areas normally exposed to heavy foot traffic patterns usually have tiles adhered the tightest. As a matter of good practice in starting the tile removal, those sections which receive least traffic should be the lcoations selected for starting the removal of the tile. Since tiles are normally in a 9" x 9" or 12" x 12" dimension, it should be the goal to remove individual tiles as a compelte unit.
6. Start the removal by carefully wedging the wall scraper in the seam of two adjoining tiles and gradually forcing the edge of one of the tiles up and away from the floor. Do not break off pieces of the tile but continue to force the balance of the tile up by working the scraper beneath the tile and exerting both a forward pressure and a twisting action on the blade to promote release of the tile from the adhesive and the floor.
7. When the first tile is removed place it, without breaking it into smaller pieces, in the heavy duty impermeable trash bag or closed impermeable container which will be used for disposal.
8. With the removal of the first tile accessibility of other tiles is improved. For the wall scraper under the exposed edge of another tile and continue to exert a prying twisting force to the scraper as it is moved under the tile until the tile releases from the floor. Again, dispose of the tile, and succeeding tiles by placing in the heavy duty bag or closed container without additional breaking.
9. Some tiles will release quite easily while other require varying degrees of force. Where the adhesive is spread heavily or is quite hard, it may prove easier to force the scraper through the thightly adhered areas by striking the scraper handle with a hammer using blows of moderate force while maintaining the scraper at a 25° or 30° angle to the floor.
10. If some areas are encountered where even the technique detailed in the previous paragraph proves to be inadequate, the removal procedure can be simplified by throughly heating the tile(s) with a hot air blower until the heat penetrates through the tile and softens the adhesive. NOTE: Handle the hot air blower carefully to avoid personal burns.
11. As small areas of subfloor are cleared of tile the adhesive remaining on the floor must be scraped up with the 4" hand scraper until only a thin, smooth film remains. In those areas where deposits are heavy of difficult to scrape the removal can be expedited by heating with the hot air blower prior to scraping. Deposit scrapings in a heavy duty impermeable trash bag or closed impermeable container.
12. As indicated in previous paragraphs, tiles should be placed immediately in a heavy duty impermeable trash bag or closed impermeable container. Do not attempt to break tiles after they are in the bag.
13. When all tiles have been removed from the floor and place in heavy duty bags or closed containers, seal the bags securely for disposal (see Sketch 18) and mark: "Caution ­ contains asbestos. Dispose in an approved landfill only."