How do I get Mother Base's budget out of the red?

I've been informed that Mother Base's GMP budget is in the red, so worker efficiency will drop, and several workers were admitted to the medical bay. I'm a little surprised this happened, since I thought my development projects were locked out if I didn't have the cash to complete them. How did I go negative? Is there a way to cut costs to bring it positive, or do I just need to get a cash award from completing a mission or deployment?

As you said, you can't assign development projects if you don't have the necessary GMP. However, there are other ways to spend money, which can get you in the red:

• Deployment: Every time you fly to Afghanistan or Central Africa, you need to spend GMP (and resources) for the equipment you'll take with you.
• Resupply: Whenever you ask for a resupply in order to restock on Fulton balloons, suppressors, tranquilizer darts, ammo, etc., you'll spend GMP and resources (thanks @JeffMercado). The price depends on your current equipment, not on how much ammo or the likes you actually need to restock on. Of course, requesting additional equipment will cost GMP and resources too.
• Helicopter support: Whether you need an airstrike, or simply picking up, deploying your chopper costs GMP.
• Extraction: Every time you use your Fulton device, you spend GMP for recovery, regardless of whether the extraction succeeds or not. People and animals cost 300 GMP. Mortars, guns, and AA cannons cost 5 000 GMP. Vehicles and Walker Gears cost 10 000 GMP.
• Helicopter replacement: Every time your helicopter is shot down, you'll spend GMP to replace it (thanks @JeffMercado).
• Buddy recovery: If D-Horse, DD, Quiet, or D-Walker ever get "killed" (or destroyed in case of D-Walker), you'll spend GMP for an emergency extraction and for treatment. If I'm not mistaken, the cost should be 10 000 GMP¹.

There are no special ways to get out of the red, other than simply collecting GMP. You can either complete missions or Side Ops (some Side Ops can be repeated), rely on your Assault Team's success during Dispatch missions, or find diamonds.

The fastest way to get out of the red would be to sell your resources. Have any vehicles, mortars, mounted guns, AA cannons, etc.? You can sell them to instantly recover in case of an emergency.

Alternatively, you can just do the target practice Side Ops on Mother Base. Avoid the R&D practice, as it's the most dangerous (you can fall to your death) and most difficult one to complete. Those Side Ops can be completed in 1 - 2 minutes, and are infinitely repeatable, though you'll need to leave Mother Base and return before you can actually repeat them.

¹: I found out when Quiet got shot down by a Soviet gunship in Afghanistan, and DD drowned by himself somewhere in Central Africa.

You can re-play through missions, and also complete side missions as they all offer a reward in GMP (as you can see in the mission selection screen, the last column on the right) also you can send out Combat Units to handle missions for you to earn GMP.

Remember to regularly send out Combat Units so you keep earning even during missions, and side missions. Another thing to keep in mind is to lower your expenses; check how much GMP your loadout costs at the mission set up screen on the bottom left.

If you want a temporary stimulation you can also sell your resources for GMP. Do keep in mind that resources then will need to be re-gathered from you or your combat units in missions. This also means if you sell vehicles, or weapons (such as the AA guns or the machine guns) then you will have to Fulton them again.

Also, if you Fulton out some vehicle, machine gun, container et cetera, then you will always land in profit by selling it. (There's no loss unless you decide not to sell what objects you Fulton out) Also Fultoning animals gives you a small profit.

Do keep in mind that selling placed weapons such as the AA gun will mean that your Mother Base will lose some of its defenses, since the ones you Fulton out are used for defense.

In addition to nolonar's excellent answer.

There are tons of raw diamonds on Mother Base that your soldiers keep hidden for themselves. CQC and interrogate your own soldiers (on any platform but the command ones) and they will reveal the locations of said diamonds. You can then collect them. Also, all the diamond re-spawn every three main missions you do thus allowing you (with some grind) to get a lot of money for nothing. Note that as a side effect, the moral will grow since you are teaching them CQC moves.

Second, you can sell resources. Some plants are worth a fortune and a quick trip into the field with D-Dog will replenish a lot of them. Vehicles are worth selling as well as long as you keep 2-3 of each. Finally, other resources (such as "precious metal") can be sold but be careful as this might limit what you can build.

This is by far the simplest way to get lots of funds if you are in the red.