If U.S. Treasuries were downgraded, then interest rates would rise, or be higher than otherwise, on U.S. Treasuries. Because the rise of rates on Treasuries would be compensation for the greater risk of holding Treasuries, it would not necessarily lead to arbitraging into other bonds and higher rates for them. There were no wide-spread ill effects on bond markets from the Standard and Poor’s downgrade of Treasuries from AAA to AA+ in August 2011.
For banks that held Treasuries, if the downgrade raised their rates it would evaporate bank equity. Whether or not it made them insolvent would depend on the extent of their Treasuries holdings and equity. Even if it made banks insolvent, the Fed would probably intervene to keep them in operation.
I don’t see that it would have an effect on the dollar, unless investors thought that the federal government would rely more heavily on monetary inflation in the future to finance its expenditures.