How do I prepare for the possibility of being mugged in the US / Caribbean?

Do not get mugged in the first place.

The following information comes from Marc McYoung and I found it very convincing and informative. The tips are also usable to avoid theft or rape.

A criminal needs (a) a victim and (b) to get near the victim

From (a) we know that if there are too many people, we are relatively safe. The danger of a safe escape is diminished, people are likely to help, identify or track the attacker, all in all it is too risky for mugging. But you are also relatively safe in the woods because there are not enough people. Criminals do not like to wait hours and hours for a victim.
So criminals will be found in what Marc called “fringe” or “transit” areas: Areas where people will pass through, but lonely and outside immediate help.
We all know such areas: Parking lots, long staircases, side roads, main roads at night, ATMs, malls outside opening times, washing centers. We are not inside the area to do something, but to get from one location to another location.

So the next step is how to identify an attacker.

In a fringe zone normal people are focused on the task to get to their location. You are searching for your car and put out the keys, you are smoking in the open staircase with another smoker or, if alone, looking bored, you are putting out your wallet to get money from the ATM, you are looking for street signs, you are preparing your laundry.
And you are going in a less or more straight way to your destination and worse, you are distracted because you are mentally switching already on your destination.
A criminal lacks all this signs because he is waiting for a victim.
You do not loiter in a transit area, you do not wait there. If you spot someone especially in the beginning or ending area of a transit zone and you cannot tell what he/she is doing there, there is something wrong. The beginning and ending is important because criminals set up traps (moving behind you to avoid escape, possibly setting you up for a pincer move with an accomplice) and they want to leave the area fast / have an excuse (I just wanted to leave/enter…).

Now to (b).

The criminal still need to get near to you and if possible, he/she wants to “interview” you (Is the victim uncautious ? Trained ? Armed ?). So if you are in a fringe zone and someone wants to get near to you: You have nothing. You have no change, you have no mobile phone to lend, you do not know the way to whatever, you cannot help because you are injured. LIE if necessary! Screw your hospitality (especially for women) – people can have hurt feelings all the time and may insult you, but it is simply bad luck that he/she
is in an fringe zone and if he/she is nice, they can wait quietly outside immediate reach.
Watch out if the person has his hands open and easily visible. If the hands slip out of your sight and stay there, something is very wrong. If someone still tries to enter your vicinity, IT IS AN ATTACK!

The other option is the typical out-of-the-bushes-assault: “Well, if my paper trail amounts to…HOLY SH..”. To avoid this, you are going on the wrong side of the sidewalk, so cars driving in your direction uses the other side. You are warned if collision is possible and the risk of a drive-by attack (stopping the car behind/before you and jump out) is greatly reduced.
If you see possible hiding places, go around them, even if you need to go in the middle of the road. (You can stay there, if the road is completely deserted and both sides are very dark).
If the street has lamps in greater distances, squeeze one eye shut in the lighted areas to retain night vision in the dark areas.

I cannot fully cite the wealth of information available, but I think that I talked about the most important points.

This is only helpful if you can call a friend for help.

I believe Western Union allows you to pick up money without an id.

You arrange a question in advance. “What is the name of your first cat”

When you arrive you give him know the transaction number, how much you sent, and what the answer to the question is.

I decided it was worth putting an anecdote in as a way to handle this, as you’ve edited now to say you’re stuck with only a couple of dollars.

In 2008 I and a friend did La Tomatina, the tomato fight near Valencia, Spain. We had a rental car, and parked at the entrance to Buñol, the town where it happens. Concerned about getting our stuff wrecked by tomato juice, we were fortunately smart enough to put everything in the car, except for a few items, which I will list:

car keys, 50 Euros cash, and mobile phone

During the fight, I was pickpocketed. When you have 10 people up against you, there’s not much you can do about it, but I was impressed given it was in a bag in my velcro-closed pocket in my shorts. Full props to them.

So now we had:

nothing

We ran into some friends there who were also there for the event, and they lent us a few Euro coins. So now we’re in the position you describe, having:

nothing, except for a few coins.

We considered breaking into our rental, but had no phone in there anyway.

Our ‘solution’ evolved, as we wandered around town. We reported it first, as this was now evidence for insurance. Next we needed to contact the rental car agency. It was nearly impossible to borrow a phone! No restaurant or anyone was prepared to help us. We used most of the few coins on a public phone calling the rental company, but they would keep cutting us off when we couldn’t speak Spanish.

Finally we returned to the Police Station, and an old cop took pity on us, and phoned on our behalf. He organised for the rental company to come fetch us, and they exchanged cars.

Next, I needed to cancel my phone. Unfortunately I was late getting to cancel that, and the weasel who stole my phone racked up several hundred pounds worth of phone bills. Vodafone refused to reimburse, despite the fact there was no way I could have contacted them any earlier. Wasn’t happy about that.

In hindsight:

• spread the money around. I was the only one looking after cash. We should have split it among us.
• my credit cards and passport were fortunately in the car. Normally I travel with the passport on me, despite what some people say, I prefer to know where it is at all times, rather than risk NOT knowing when it’s gone, at least this way I will know when it’s gone wrong.
• photocopy everything.
• cancel everything as early as you possibly can. Credit card companies, phone companies will often try their best not to refund you.
• have a lock on your phone. I did, but they used the sim in another phone. However, it at least requires they have another phone to transfer to, which saves you at least a minute of them using it.
• when possible, keep cards separate – maybe a spare credit card in your sock, for example.
• always keep an eye on your stuff. It doesn’t take long to develop a habit of just glancing at your bag, or subconsciously feeling in your pocket.

Wikihow has a piece on this and suggests:

• Drab down. Expensive clothing’s purpose is to advertise to the general public that you are filthy stinking rich. Robbers don’t have
  X-ray vision: they notice the attire only and guess what’s in the
  wallet by that.

• Buff up! If you were a thief, would you go for someone wimpy or ripped? You’d go for the person less able to defend themselves and
  avoid a nasty injury. Not only does working out make you appear harder
  to rob, it makes you actually harder to rob.

• Cover up that bling. Or, don’t wear it at all. Think about it: unless you’re on your way to a presentation or job interview, nobody
  cares about your $500 watch or $200 Paris Hiltons. Nobody, that is,
  except those with bad intentions. Save your valuables for momentous
  occasions or for times when there is less of a risk of getting
  robbed–when you’re with a large group of close friends, for example.
  If you absolutely must–cover up that watch. Put on your new pearl
  earrings after you get to your destination. Nothing says “rob me” like
  valuables in plain sight.

• Keep a “robber’s wallet.” Put high value bills/notes in small plastic bag in your sock or some other safe place. Buy a cheap wallet
  and keep some low value currency and some fake cards in there–to make
  it seem less like the fake it is. If you get robbed just hand the
  wallet over.

• Don’t flash the cash. This is even worse than displaying valuables! If you pull a $20 out of your pocket and then another $20 later,
  thieves will assume there’s more where it came from.

• Look sure of yourself, even if you aren’t. Just exuding confidence tells a thief that you’re liable to fight to get your money back.

I agree with @Kendell’s suggestions, and here are my additions:

1. Don’t keep your (all) credit cards in the wallet. I worked in a bank once, and the security there said that most people carry everything (licenses, ID’s, cards, checks) in one wallet, and when it gets lost or stolen their whole life may be ruined. Keep 1-2 cards in the wallet, another one somewhere else on you, and all the rest leave at home/hotel safe deposit box.

2. Never carry checks with you.

3. Never carry Social Security card with you.

4. Have a spare ID (for example – your driver license is in the wallet? Keep your State ID/Green Card/Passport Card/Military ID somewhere else). This way you can relatively easy prove your identity to the airline/airport security when you’re trying to get back home, or to the consulate when you’re trying to get a temporary passport.

5. Several banks/credit card companies provide a service where you give them all your credit card numbers, and if you report them you lost theirs, they’ll take care of reporting to all the rest. This way you only need to make one call and won’t forget anything. I know AMEX has this service, for example.

6. If you have visas in your passport – make sure to save copies so that you can recover them easily later instead of reapplying. Generally, don’t carry your passport with you when you travel, keep it in the safe deposit box (laws in many countries require you to carry it with you, but other than Russia, I haven’t heard of any country actively enforcing it).

The best approach I think is to not worry about being mugged, but to make sure being mugged will not be a disaster if it does happen:

1) Always travel with multiple credit cards, only carry one with you out of the room.

2) Always keep photographs on a smartphone or something else of all your cards and the backs of them (so you have the numbers to call to report them stolen).

3) Don’t carry any more with you than you absolutely need – when abroad for example do you really need your driver’s license, costco card, etc? The less you have the less they can take.

4) If you get a lot of cash for the trip do not carry it all with you all at once.

5) Always copy photos off your camera or smartphone, or switch out memory cards every night so that your camera being stolen means only a days lost photos (using the camera connection kit to copy photos into an iPad every night is a great way to back them up without having to take a whole computer along).