Advertise this service opportunity at your church.
Decide how long and how often you be doing this ministry. Consider doing it in the weekend mornings (11am-2pm) or evenings (5pm-8pm) when traffic may be higher. If you can do it for a few consecutive weeks, you might be able to build relationships with the regulars there.
Collect supplies. You will plenty of quarters. Each load at the laundromat can cost $0.75-$1.50. Ask your congregation to help fund this outreach. Set a place in your church to collect weekly pocket change and/or laundry detergent. Other supplies you may need are:
Crayons and coloring sheets
Grab bag with toys for children
Laundry detergent
Dryer Sheets
Refreshments like baked goods and coffee
Materials about your church for those who are interested
Seek permission from the owner of the laundromat to set up your outreach there for a day. It is also helpful to ask the owner about who the regular clients are, as you may need to recruit Spanish speakers.
At the laundromat, greet people as they come in. Offer to pay for the laundry and fold laundry. Have the supply of detergent and dryer sheets readily available.
Spend time with those who are open to talking. Find out their needs and ask if they would like you to pray with/for them.
Tips
Wear shirts that can identify you and the volunteers as a group.
You can even bring some of your own laundry to do. Some people may perceive you as looking down on them, and this helps you identify with the people there.
Be careful about giving out your personal contact information.