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Ok, this should be an easy fix for you to perform!!
On the topside of the washer just under the counter there should be 2 small metal brackets that stick out just enough to get a screw into the hole that is in the bracket.
There are the brackets to secure the top of the washer to yhe counter that will keep it from tipping.
If you have a counter that has wood on the underside then just take 2 small (short) screws and screw the brackets to the counter. If you have granit or marble you will need to install it differently. In this case you will need to either drill 2 very small holes in the counter (at the bottom side) and install the screws.
That should take care of the issue for you!!
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Im not sure you can get them that small. They need to be less than 1/2" so you dont crack the granit. If there is a gap of about 1/2" between the counter and the washer bracketts then I would glue a piece of wood to the counter and then use the wood screws into the wood. This way you dont damage the counter by mistake.
There is normally a lip that will hang over the counter to hide the lip where the counter sits on the cabinets, you can glue the wood (like a piece of 1/2" plywood or other wood cut to fit) to the counter just behind that lip with liquide nails and that will hold great.