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If an egg is already cracked then it will almost certainly split open when boiled, there's no way to avoid it, it would be best to cook them using another method. Be aware that once eggs are cracked they no longer have protection against microbes, if you don't know how long they have been cracked it may be best to throw them away.

I use Egglettes™ to boil my eggs. You simply crack the egg/s into a silicon cup and place it/them into boiling water. Without wishing to seem like a crappy infomercial (see below), I find it a heck of a lot easier to crack an egg into a cup than I find it to peel eggshells off of a soft-boiled egg.

Obviously it makes no difference if the egg has already sustained a recent crack (from being dropped onto the surface, for example) since you're removing the shell prior to cooking.

https://www.youtube.com/watch?v=WxVT2Q6Vt_c

You can choose to poach the egg.

The egg is cracked into a cup or bowl of any size, and then gently slid into a pan of water at approximately 75 Celsius (167 °F) and cooked until the egg white has mostly solidified, but the yolk remains soft. The "perfect" poached egg has a runny yolk, with a hardening crust and no raw white remaining.

I don't have any experience with this but I would imagine you could cook it until the yolk hardens along with the whites.

Adding salt in the water helps as well, this increases the egg white will become solid when getting in contact with the salt water and basically it will close the crack again.

Believe it or not, you can put some Duct Tape (like gaffer tape - heavy tape) over the crack, and away you go.

(I'm not sure why anyone would bother with this - boiled eggs which have some white "poking out" are fine - but there you have it.)

It's possible to boil an egg with steam. This usually requires an additional device. The eggs are put upright and won't split open since they are not exposed directly to the hot water.

If the egg's been cracked for a while, just throw it out. Even though the boiling process should kill anything growing in it, good sense says it's not worth the risk. If the egg gets small cracks when you put it in the water or while cooking, it's perfectly fine. In my experience, the heat will quickly make the outer layer of white solidify as it cooks, which will lock everything else inside, crack or no. The worst I've ever seen happen from cooking an egg that cracked early on is that little bits of white escape and get cooked in the water, giving it a cloudy look. I've eaten many eggs that cracked while cooking and never had any problems.