It depends. You might first want to review this IRS website regarding gift basis (which determines how much gain/loss you report): IRS cost basis. Per the website, your father's cost basis really would only come into play if, at the time he gifted you the coins, they had lost value from when he purchased them originally. If that happens, your gains would be based off of your father's purchase price of the coin, and loss would be determined by the fair market value of the coin when you received the gift. If the coin either maintains or increases in value, your gain/loss is then determined off of your father's cost basis when purchased.
With that as a background, if your father is directly purchasing stable coin, and transferring it to you in that format, you will need to report your father's cost basis upon purchase of the coin. You do not need to submit specific proof of this basis unless the IRS asks for it (for example, in an examination or audit). However, if you do need to prove it, the easiest way is to maintain a record of the date of acquisition of the coin, and it's value at the time. This is what the IRS will use if they need proof. Although there is talk of legislation requiring crypto-exchanges to more thoroughly report activities, I would anticipate that the majority of exchanges are still not reporting on any official forms, so it is necessary to manually track your transactions. Still, with a stable coin, you are correct that you should not see significant movement, and therefore very little if any capital gains or losses. But you will still need to report the activity via Forms 8949 and Schedule D, so it is still important to track your transactions.
If your father purchases another type of coin, however, and you wish to transfer that coin into a stable coin, this is a separate transaction that very well could result in gain or loss, and would need to be reported separately and in addition to any further sales of the stable coin. (In this scenario, the value of the stable coin is your purchase price after having exchanged the prior coin).
@Opus 17 Thank you for the clarification, and I have modified response as a result
[Edited 12/6/21 6:41 PST]
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