You can never change just one part on a bicycle. Everything on a bicycle works together. Almost anything that you do will make it necessary to change some other part to make the whole package work.

If you have a bike with down tube shifters, it probably has 126 mm or even 120 mm dropout spacing. Since most STI shifters are for 8-speed or more, you'll probably have to adjust your frame spacing to accommodate a wider rear hub. What size tires do you have? Lots of older bikes have 27" rims. New 8+ rear wheels are going to have 700c rims. You might be able to adjust your brakes down to fit the slightly smaller wheel size, but I wouldn't bet on it. Are your existing shifters friction or indexed? If they're not indexed, you might as well put new derailleurs on your shopping list. Finally, STI shifters are probably the most expensive component on your bicycle. They're not super problem prone but, the grease in used ones that have been sitting around for a long time is likely to have dried up so they might not be reilable.

Bottom line: It definitely can be done but the cost will vary with how much you have to change.