How can I prevent my lawnmower from spreading red thatch?
1 cup (5% laundry) bleach to 1 gallon water is usually a good bet to kill most things. Use in a well ventilated area, and never mix ammonia and bleach. (Which you all should know already, but evidently some people still don't...)

With a lawnmower, getting the blade and underside clean enough to paint (remove all grass clippings - and hey, if you've gone that far, why not paint it!) will help immensely in not having residual clumps of stuff that might not have gotten bleached all the way through. Clean (but don't paint) the wheels/tires (if any) as well (evidently hover mowers are still out there, wheel-free.)

"Red Thread" seems to be a more common name (I get no results for "red thatch" other than "red thread" and "pink patch" results.) Both are fungal diseases. A bleach solution will kill fungus on non-porous surfaces (such as your mower's steel and plastic) but is less effective on porous surfaces (which would mostly be the "age of the mower old" buildup of impacted grass clippings, in lawnmowers.) 

Sharpening the blade is also a good idea for prevention, as these diseases more easily attack damaged grass blades than ones that have been cut cleanly.