How can I save the world from the spread of Hallow and Corruption/Crimson?

With the Clentaminator present in the game, this is actually quite easy, if a bit time consuming and expensive. Dig hellevators every 120 blocks across the area you want to purify. With the Green Solution equipped, float down the each hellevator twice, first spraying to the left, then to the right. I recently did this in one of my worlds, and it worked perfectly. I'd recommend checking every so often after you're done to make sure you didn't miss anything. This will also turn any underground mushroom biome to jungle, so you may want to go back when you're done and convert it back with Dark Blue Solution.

When I did this, I left large swathes of land from the surface to the underworld converted, and even brought in Crimson from another world so I had all the biomes present. Just make sure that if you do this you put wider shafts (I left six blocks) around the corrupted/crimsoned/hallowed areas so they don't spread.

Ever since the 1.2 update, the NPC steampunker (gained by killing a mechanical boss) sells the Clentaminator (for 2 platinum coins).

This used with green solution (25 silver) as the ammo will return anything it hits back to the purity (normal grasslands).

It has a range of about 40 so you can make tunnels every 80 blocks and spray in both directions.

Pre-hardmode, your best bet is probably to limit the 'inconvenience' of Corrupted turf, and latter surgically remove areas of severe risk, before summoning and defeating the Wall of Flesh boss (activating Hardmode and the rapid spread of corruption/Hallow).

What I do is to chop down all trees (and remove any chests sitting on grass) in a corrupted area, and then beat the tar out of any and all grass (and thorns) on any exposed surface. I then brick over the entrances to the Chasm, and sprinkle Purification Powder on exposed Ebonstone (Vile/Purification Powders normalize, Holy/Unholy Waters convert a target to the extremes). What ends up happening is that generally, foot traffic through these 'tame' Corrupt areas tends to get interrupted much less by Corruption beasts (with the exception of the occasional Devourer), as most of the "Eater" creatures tend to have to come from the bottom edge of the screen, and get trapped inside their pits.

At this point, I mainly visit the Chasm for Shadow Orb breaking, and collecting Vile Mushrooms (and ripe Deathweed during Blood Moon, until I get enough seeds for a farm). On each visit, I dust the walls/floors with a stack or two of powder. Eventually, when it looks like most of the surface area in a section is pure, I dig down through the pit wall or chasm floor/ceiling, looking for any remaining ebonstone to nix, or corrupt grass spots to pound out. I flatten the surface to make fighting summoned Eater of Worlds bosses easy.

I advise that if you go about 'removing' Corruption, you take your time and do it before you go Hardmode, so that you don't have to do it all in one long, tedious run to avoid a relapse. Even then, I haven't really played into Hardmode yet, so I can't say how drastically you'll have to revise this strategy when it's triggered.

Generally speaking, there's enough clay or unexposed dirt in most strata of the underground's upper layers that you needn't fear needing to blow up the whole world. Remove corrupted/hallowed grass from as much of the surface (and surface caves/pools) as you can, and you should be fine.