All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Dealing in US dollars or any other currency is permissible under Sharee‘ah. The Prophet, sallallaahu ‘alayhi wa sallam, and the Rightly-Guided Caliphs used to deal in dinars and dirhams while these currencies belonged to the Persians and the Romans. This dealing continued to exist until the Umayyad Caliphate where they coined the Islamic currency. Such a matter should be done in accordance with the exchange rules regarding different currencies.
It should be known that a Muslim is rewarded for abandoning this matter when there is a good intention behind it, such as not to help those who support the Muslims' enemies to have their currency spread and have its value increased.
Allaah Knows best.