Are the US's promises to make payments under the Social Security
program constitutionally protected?
The United States government has made no such promise (prospectively anyway).
If the law in effect on a particular day says you are entitled to a Social Security payment on that particular day, you probably have a debt owed to you by the United States on that day if the debt is not paid (e.g. due to a computer glitch), because at that point it becomes a vested property right which you could bring suit in the U.S. Court of Claims to have paid to you.
But, you have no vested property right in future payments from Social Security that are not yet due under the law.
Please consider 3 cases:
Just before the trust fund runs out, Congress passes a law cutting
benefits effective immediately.
Congress can change benefits at any time. Social Security benefits are not a debt (as noted by @dsolimano in another answer to this question).
The Treasury bonds in the Social Security trust fund may very well be a debt, but because the Social Security Trust fund is as a matter law of law entirely invested in U.S. Treasury Bonds it is matched by an equal and offsetting asset. It is money that a fund owned by the United States is owed by the United States. It is not owed "in trust" for the beneficiaries of Social Security, it is owed "in trust" for the United States government's Social Security program.
Congress does nothing about Social Security and the trust fund runs
I assume that you are saying that this is an issue if Social Security payments stopped being made because the Trust Fund ran out.
Incidentally, I don't think that Social Security payments to beneficiaries would stop being made if the trust fund ran out of money, because, as an entitlement, Social Security payments don't need annual appropriations to justify them being made by the U.S. Treasury. The trust fund balance would merely go from positive to negative, causing the Social Security program to owe a debt to the United States government in general.
Also, if the trust fund runs out then the there would be no debt to dishonor because all funds in the trust fund have been paid out. Again, the debt, if any, is not the right to Social Security payments. Social Security payments are a non-vested discretionary payment made to beneficiaries pursuant to law, not a debt, until the day that they become payable to a particular person on that date.
Government shutdown way before the trust fund runs out (e.g. over
border wall funding)
First, I don't think that Social Security payments would stop in the event of a government shutdown because as an entitlement Social Security payments do not require regular Congressional authorization and government shutdowns happen because there are no funds appropriated by Congress for certain functions. Social Security payments have not stopped in these circumstances in the past.
Also, to the extent that payments stopped going out because, for example, no one showed up to the work to process them, the Social Security payments would continue to be owed and the Treasury bond debts owed to the trust fund would not be dishonored or depleted in any way.