How can I connect a 120V washer to a 240V circuit?
You're going to have to either get a washer that accepts 240V, or change the circuit.
New Washer
Getting a new washer will depend on the manufacturer, dealer, and other factors that are off topic here.
Change the circuit
Modifying the circuit will require knowledge of electrical safety practices, and general electrical wiring principles/standards/codes. If you lack this knowledge, or are not comfortable working with electricity, please contact a local licensed Electrician.
If you decide to tackle this project yourself, here is an overview of what needs to be done. You'll be going from a configuration with two ungrounded (hot) conductors, and one grounding conductor (NEMA 6-20). To a configuration with one ungrounded (hot) conductor, one grounded (neutral) conductor, and one grounding conductor (NEMA 5-20).  To do this you'll have to swap out the double pole breaker for a single pole breaker, and repurpose one of the conductors.
At the Panel

Before you begin, you'll need to purchase a 20A single pole circuit breaker that is compatible with your service panel, and a blank panel cover insert.
Start by turning the power off by flipping the main breaker to the OFF position, then verifying that power is off using a non-contact voltage tester.
Remove the service panel cover (be careful working in the open panel, the main lugs are still energized even when the main breaker is in the OFF position).
Locate the breaker for the washer.
Remove the wires connected to the breaker (should be either black &amp; red, or black &amp; white).
Remove the breaker from the panel.
Install the new single pole breaker in one of the slots where the old double pole breaker was.
Install the blank panel cover insert in the panel cover, so that is covers the empty space left.
Connect the black wire that you removed from the breaker in step 5, to the new breaker.
If the other wire from step 5 is white, connect it to the grounded (neutral) bus bar. If the wire is red, mark the wire with a bit of white tape, or a white marker, then attach it to the bus bar.
Reinstall the panel cover.

At the receptacle

Before you begin, you'll have to purchase a 125V 20A receptacle.
Uninstall the old receptacle.
Connect the black wire to the brass screw terminal on the new receptacle.
If the other wire is white, attach it to the silver screw terminal on the new receptacle. If the other wire is red, mark it with white tape or a white marker, and attach it to the silver screw terminal on the new receptacle.
Attach the bare or green grounding wire to the green grounding screw on the new receptacle.
Mount the receptacle in the box, and install the cover plate.

Once everything is finished, you can flip the main breaker, and the new breaker to the ON position.
WARNING:
This answer is only applicable in the US, and possibly Canada.