How do I build the Beacon in a Swamp

Yes, swamp will slow down followers and get them stuck, and sometimes kill them. The longer they spend on the swamp (e.g. working on the beacon) the more likely they are to die from it, or get tired and return home.

To clear out the swamp, use the beautify god power. When using beautify, make sure you tap rather than drag, as it uses a lot of belief and you can get the same effect for a lot less belief by placing it in "dots".

On the android version, beautify is awarded when you have 30 farms.

Beautify

As Samthere points out, the Beautify power will permanently remove swamps/blighted land and solve your problem. However, there is a swamp covered beacon fairly early on in the ios version of Godus and most likely you will not have sufficient population/mines/fields to unlock the Beautify card yet, nor would you be able to find enough stickers to activate the card if it was unlocked.

Let Followers Die for the Cause

Instead of clearing the swamps out so that your followers survive, you will need to send followers to work on the beacon until they die, then send more. This micromanagement makes this one of the more frustrating beacons to work on, but it can be done. I recommend leashing 8 followers at once, sending them up to die while they work on the beacon, then sending another team of 8. The followers won't even survive long enough to get tired and stumble home. Slowly but surely, you will complete the beacon and progress further in the game.

Will Boosting Keep Followers Alive Longer in Swamps?

No, boosting the followers while they are in the swamp does not make them survive any longer. I have seen followers die while working on this beacon even at full stamina from a recent boost. I suspect that simply being in the swamp for a certain length of time kills the followers and boosting has no effect.