A security deposit, classically, is money for safekeeping (pikadon), whose purpose is to protect the landlord’s interests with regard to the property. It is intended to be returned in full, if everything ends without incident. If given by a check that is not cashed, indeed nothing of significance happened. If given by cash, the landlord, at least conceptually, must have it ready to be returned when appropriate.
The mishna[1] says that if Reuven gives money to Shimon to watch, Shimon may use it for his own purposes (and return other coins) only if he is a money changer and, even then, not if Reuven demonstrates that he wants the coins to remain intact. The explanation is that a money changer is in the practice of using any money that is available to him for making exchanges, but others are assumed to literally hold the money on behalf of the owner. Thus, assuming you expressed no interest in using the money you were given, it might be argued that you were expected to have kept it, to return the same bills you received, and not to use them for your purposes. In that event, you have to pay at least the value of what you took at the time you took it[2] according to the local currency, which, in Israel, is the shekel. Thus, you would pay back the amount of shekels you received, irrespective of the value of $1,000.
On the other hand, it is possible that in modern times, we treat the standard person like a money changer in this regard,[3] especially where people are reluctant to keep cash in safes. When one uses pikadon money with permission, he becomes fully responsible for it. The Shach[4] views the use of such money as a loan, not like borrowing an object (she’eila). One difference between the two categories is that if the currency goes out of circulation, a loan must be repaid with a valid currency, whereas in the case of she’eila, one can make the return with the (now invalid) currency he received. Another is that a loan makes this question not only one of monetary rights, but also of potential ribbit.[5]
You might make a similar argument. You took a loan of $1,000, the sum mentioned in the contract, and it is $1,000 that you should return – no more and no less – regardless of the bills you received or will return. The laws of ribbit dictate that if one borrows a certain commodity, it is forbidden to stipulate that he must return the same amount of the commodity even if the commodity goes up in price.[6] However, one can lend a certain amount of currency and demand the same amount of the same currency in return, even if it appreciates in value in the interim, as long as it is the recognized currency of the place, not foreign currency.[7] Although that would seem to indicate that in Israel, only shekels can be used as currency for matters of ribbit, there is a decades-old halachic assumption that the dollar’s special status in Israel makes it equivalent to the shekel in this regard.[8] (It is questionable whether in the present financial situation in Israel this is still true, but when you made the agreement it was.[9])
However, unless your contract is unusual, the above is not relevant. Generally, the designation of US dollars determines the amount of shekels to be given when payment is due or paid. It is simply a pricing tool. However, the payment in Israel is still ordinarily in and of shekels, and thus the security deposit was, predictably, in shekels. Therefore, even if we decide to look at the deposit as a loan (which is questionable),[10] it is a shekel loan to be returned in shekels or their equivalent, unless specified otherwise. Had the dollar gone up (as it did for years), your renter could not have demanded that you return more shekels than you received. Similarly, now that it went down, you may not return fewer shekels than you received, even if you personally view your finances in terms of dollars, and there is also no problem of ribbit in so doing.