How did people deal with checks before computers and other things that made it easier to verify?
The same way they handled the precursors to checks, which were promises to pay. Kind of like running a tab at the bar, that privilege is reserved for trusted parties or those so well known that the hit to their reputation would be more damaging financially than the amount being floated.
Imagine you live in a area where everyone knows where everyone else lives. We know your address, what job you have, and roughly how well you're doing financially. There's no such thing as checks. You stop by the store, want to purchase some items and won't be coming into "town" for another week, but aren't carrying negotiable instruments with you and don't have the time to get them for the purchase now. What happens? Well, if you're well known, with strong ties to the community, and have been honest in your dealings in the past, and the store can tolerate not having payment for a week or two, the store may decide to record your debt in their books with the expectation that you'd pay it later. If you didn't pay, you may expect a visit later from the authorities or you'd find yourself blacklisted not only from the merchant you stiffed, but using such services at a wide range of other merchants.
So, given that you need to live and trade with these people, you are established in the community, moving somewhere new would be problematic, not to mention much more expensive than whatever it was you wanted to buy, why would you skip out on the debt? Or even take it on if you had a strong likelihood of not being able to cover it?
But this system is problematic. It requires you to physically come back to the town with the money (potentially dangerous) or go to the bank, withdraw the funds, then pay the merchant (time consuming). And the merchant is essentially giving everyone who takes advantage of this an interest free short term loan (not necessarily good business).
The next month, the banks announce a new innovation - the "check". Which is now a piece of paper that when presented to the bank, results in a transfer of funds from the payer's account to the payee's. This reduces the amount of time the merchant needs to float funds from days/weeks/more to just business days. This removes the need for the customer to return to town, which saves them time. But it changes nothing else. You still only conduct such transactions with people you trust or people you can leverage into paying up.
If anything, the ability to defer trust to an electronic system is likely what enabled even more fraud. Since you don't need to exploit personal connections and trust, the threshold to scam someone is much lower.
So, TLDR - this worked due to tight communities and some amount of trust which predates the invention of checks. For you to pass a fake check, you needed to be able to execute a confidence scam - convincing someone you were trustworthy or person of importance/reputation, or convince someone who already has those qualities to either vouch for you or put up their own money while accepting your worthless check.