Does a permanent resident have to pay taxes on foreign family gifts?

I am going to reply in generalities since I handle US taxes principally from the stand-point of resident in a foreign country. But most countries follow a generalized scheme, so you are likely to find that it is the same for you. Just verify please.

1 - Gifts are gifts. US is unusual in that they tax gifts, but they tax it in the hands of the giver, not the recipient. You should not be paying taxes on that. 2 - You are required to declare foreign holding in certain cases. The form is TD F 90-22.1 and it is filed separately from your tax return. Check the instructions. 3 - In most countries, you are required to declare your holdings at the time of immigration since you would be owing taxes to the country where you left and you will owe taxes from that point on to the country in which you arrived. A number of countries allow you to defer the payment of the departing taxes until such a time that you divest yourself of the holding(s), though security may be occasionally required. This is principally items reported on Schedule D on US tax returns, though you may also have issues with asset depreciation.

Are your family non US citizens and not US residents? If so, then: No. But you do have to report the gifts if they are above a threshold, which is something like 90k or 100k. See form 3520 and instructions. The penalties for not reporting are steep, essentially like a 35% tax.

Technically one must pay capital gains tax on the increase in value if the increase is above $200 (or $100, I forget). You'd have to calculate the capital gain on each amount of interest. Say you originally put in $5000 when the exchange rate was x1, then got 0.25 EUR interest when the exchange rate was x2. Well know, convert each bit to US dollars, and pay tax on the capital gain. There is no capital loss allowed. But I wonder what a practical person can do.