I've cleaned a lot of engine bays and here are my observations:
1. Only use a hose with a nozzle that sprays. You don't want power washing here, you also don't want such large amounts of water to pour out like out of an unrestricted opening of a garden hose or pouring water from a jug or anything like that. Both are too much.
2. Before starting, find all of the places that could be potential problem places and check them or seal them. Here are some examples:
* Under hood fuse boxes: Check that the rubber seal on the lids are still working.
* All fluid reservoirs: Check that the caps on the fluid reservoirs are all sealed.
* Dipsticks: There have been cases in the past where dipsticks were not water tight. You'd be surprised at how bad of a seal some dipsticks have. I don't take chances here with a hose when I can just go over it manually with a wet towel and not worry about it. Honda has had some really loose dipsticks and Chrysler had a recall on some of their automatic transmissions because a bad dipstick seal allowed water contamination. Find the location of the dipsticks and avoid these areas entirely (tape a shopping bag around the top or something).
* Intake piping: If you have a vacuum leak your engine would probably run like garbage but there have been cases where people had tears in their intake snorkel past the MAF and didn't even know it. Along the same vein make sure the lid on your air cleaner box is properly secured.
3. I wouldn't use a degreaser, only water but that's just me. The only time I would use a degreaser is if there was serious oil to clean up and then I'd be very careful about where to spray it. I've used degreasers in the past and have had problems arise. One example was where a little degreaser mixed with some water ended up past a plug wire boot seal and the spark plugs ended up completely corroded (as in ashy white).
4. If you hose a hose spray like I mentioned above, electrical connectors and even the alternator are safe to skim by with the spray, as long as you don't focus on them for any length of time.
5. I usually use the hose only to wet down the engine and hose off any loose dust, dirt, salt or debris, then I use a wet microfiber towel (along with water in a bucket) to manually wipe down all of the areas and physically clean the dirt off. The areas you didn't hose off like dipsticks are handled manually with the towel.
6. Keep moving the hose around, don't flood any areas. Be mindful of water possibly building up and ended up going past reservoir cap seals for example. Also be mindful that some things like dipsticks could be in recessed places where water could build up and force their way past the seal. For example some Honda AT's have dipsticks that are on the transmission case itself, not a tube attached to the case. On some of these it is possible to have water build up around the recessed dipstick if you're not careful.
7. Wring out the microfiber towel so that it's damp but not soaked and use it to soak up and dry all of the areas that you went over wet.
8. If you have an air compressor (pneumatic or electric), use it to blow the air out of the engine bay to speed up drying. This is optional.