Here are a couple of pointers:

1. Don’t go by what the print or initially state as ID requirements. Ask to see the manager. Oftentimes there’s a lot of discretion the manager has in what to accept as security. It is very likely that they are willing to accept some other form of ID as security outside of what was stated as the default instruction.
2. In most situations the renting-business is ideally sufficiently protected against loss or theft if they can take a hold (I forget the technical term for this) on your credit card. This is what hotels etc. do all the time. That way if you default they can charge the card the amount. You can always offer them that option if they are willing to be flexible.
3. Offer to keep a cash deposit. I’d rather lose cash than my Passport.
4. Keep with you a wide array of authoritative looking acceptable ID. Some options: A personalized Photo Credit Card, A Work ID, A Student ID, A Youth Hostel ID, International Drivers License with translation, Frequent Flyer Cards if they come with Photos. It is amazing what kinds of IDs you can get away with as security just so long as they look professional enough. I often use those at iffy places where I’m not too confident they won’t lose my ID.
5. Offer to leave multiple IDs behind instead of a Passport.
6. I had a US-State ID before I got my drivers license. I used to use that everywhere in the US because it looks almost identical to a driver’s license. And I will have much more trouble if my driver’s license gets lost.
7. Some states let you keep the old / probationary drivers license after renewal. Use that as a disposable ID you can afford to lose. (Well, it does raise the option of subsequent misuse but that’s different)
8. I have my old, expired passport with me. Although the issuing office takes real pains to make it evident that it has been cancelled it is amazing how many clerks will not even notice it when offered as ID.
9. Finally, if everything else fails, take your business elsewhere. It really is too much trouble getting a new passport if it gets stolen.

1. Make a hi-fi copy of your passport, say color.
2. Repeat step 1 a few times.
3. Anyone who asks to see your passport ( for ID, collateral purposes ), offer them one of your hi-fi copies, and let them sight ( and not keep ) your original.
4. If that is not good enough for them, do as other answers suggest and take your patronage to more civilized climes.

If people have your passport they effectively have leverage over you, not only in terms of it costing you a couple of hundred USD to replace it, and also in terms of impersonating your identity or framing you.

If you like this idea, maybe do it with your other state issued ident documents so you have a whole carpet book of copies of various IDs, the originals which you keep and can produce for sighting purposes.

Really, only border security and related functions should absolutely require your original passport, everyone else should be okay with a original-sighted-copy. If they’re not, they likely have too much leverage on you and you have introduced an unwanted risk to your adventures by providing your passport to them.

I lived in Korea for four and a half years. While there, I occasionally encountered places that asked for my passport, but I never carried it with me. They always accepted an ID card instead.

In general, rules in Korea aren’t enforced terribly strictly. Most things are negotiable. If you give a believable reason why you can’t provide your passport (can’t, not don’t want to), odds are you’ll be able to negotiate some other way. (And any service aimed at foreigners will have English-speaking staff. Or, call 1332 (02-1332 from a cell phone) for translation.) A reason as simple as “I don’t have my passport with me” is probably good enough.

By the way, under Korean law, foreigners must carry their passport or alien registration card at all times and show it to the police on request. However, I’ve never heard of the police asking to see a passport or ID card; if you don’t look Korean, the police will assume that you don’t speak Korean and generally won’t want to speak English, so they’ll leave you alone.

Edit

If you decide to leave your passport as requested, it’s highly unlikely that there will be a problem. In all the time I lived in Korea, I never experienced anything to make me think that there would be any significant risk. ID theft is quite uncommon. I lost my wallet on the bus once and it was delivered to my apartment by the police a few hours later. In my opinion, you’re much less likely to be a victim of crime in Korea than in most other parts of the world. If it was me and I had my passport on my and not my alien registration card, I wouldn’t hesitate to give them my passport.

The sensible thing to do is to

take your business elsewhere.

I understand the need to hand over the passport.

There is no need. Your passport does not belong to you (it belongs to your government), so it is not appropriate for anybody to use it as security against a loan or rental. If an organization insists on using a passport as anything other than an identity document, you should take your business elsewhere.

I wonder whether leaving a passport card would be acceptable. On a recent trip to Jamaica, we flew in so had to use our book passports, but we each also have passport cards, and we had them with us. I’d have hated for the passport card to have been stolen or lost, but even if it had, we’d have still had our book passports, so no issues as far as when we flew home.

As a hack, you could buy an imitation/”novelty” passport (maybe in the name of an imaginary country), which the average bike-rental attendant may not be able to identify, and which doesn’t carry any risk to you of being misused.

Using such a passport to cross a border or as identification to officials or police is doubtless illegal and a bad idea, but for such a commercial enterprise, it may not be*.

*not legal advice – for other information that is also not legal advice, check the wikipedia page.

Depends on your country, but I usually handle such situations using my official government-issued ID card instead of passport. It’s official enough for many places, however still not as useful as passport for eventual criminal activity.
And, more importantly, you still keep you passport so you won’t risk problems with going back home.

Don’t use them. It’s really that simple.

The only people who need to see your passport are immigration and border control, or law enforcement. (ETA: in some jurisdictions, businesses such as hotels or car rental agencies are required by law enforcement to keep a copy of passport details on hand)

If you’re asked to submit your passport to confirm your ID, that’s acceptable so long as you don’t allow it to leave your hand, or if at a bank or similar, passing over the counter on the understanding it never leaves your sight. Submitting your passport as a kind of commercial insurance is unethical, and places the passport holder at considerable risk if they make contact with law enforcement and cannot show alternative acceptable ID. (E.g., in such places as the US state of Arizona with controversial “show your papers” laws in effect, law enforcement may detain such a person until such time as they can produce immigration or travel documentation.)

Even if the operation seems to be trustworthy, you’re a visitor – you don’t really know. The consequences of a lost or stolen passport are forfeited tickets, and unplanned overstay. The latter may lead to an illegal immigration status, which may place future travel or immigration plans at risk. While the risks of those occurring are reasonably low, the costs of such consequences are far too great to ask for such a low cost service as bike rental (when there’s already a cover cost!) or self-guided audio-tour rental.

Protection sometimes means complete avoidance. Often (as others have noted) there’s an alternative means of leaving a deposit; or there could be an alternative means of acquiring the same or similar level of service (for example, outright buying a used bike and selling it on).

On a trip to Amsterdam I had the same problem trying to hire a bike and I decided not to handle my passport. If they lose or misplace your passport, you are on your own, in a foreign country without a passport. I would be more in peace of mind if they hold my credit card than my passport, but they would not take it as safeguard so, no bike ride.

Do you really have to leave a passport? Would a credit card suffice instead?

In Istanbul, to rent an audio guide at many public attractions, they often want you to leave a passport or identity document, or failing that a credit card, as security. This is to encourage you to return the audio equipment at the end of the tour. I haven’t actually seen this outside of Istanbul, but it made me nervous, just as in Seoul you were nervous about the loss of your passport.

So, each time, I left a totally worthless loyalty card instead—the kind of card you wouldn’t even bother to get replaced if you lost it. It was accepted everywhere I went. (It was a Flying Blue gold card, I was a bit disappointed no one stole it.)

You already cover the sensible things that I would also do, especially online reviews and making a copy. Perhaps you could take a photograph of the person you handed it to? If you wish to go further, I have one possible suggestion:

I recently bought a bike and took it for a test ride. To do so, I gave them my credit card and driving licence which they put in a lockable box and gave me the key. Obviously they’d have some other way of opening it if I didn’t come back, but at least it might be restricted to the manager rather than anyone working there.

It’s unlikely that you’re going to find a similar mechanism in Seoul, but it does at least provide you with the idea that you could buy some sort of lockable or at least tamper evident box or seal, show them your passport, seal it in the container and give it to them. They could break it open if you don’t return. If you do return, you could see that they hadn’t tampered or copied your passport.

Tamper evident seals are cheaply available on Amazon and you could just seal any old plastic box. Obviously this does nothing to prevent them stealing it though.