What is the best way to clean lattice planters?

You could use a power washer or use an attachment for your hose that will give you more water pressure. See here for more information: https://www.hunker.com/13404448/how-to-turn-your-garden-hose-into-a-pressure-hose

I don't exactly have lattice on my large square plastic planter boxes but they collect dirt in the corners that is hard to get out. I find that using a "sponge flower" on a handle works well to get soapy water into all the nooks, crannies and crevices and then a good rinse out with the hose. This is an example of a sponge flower on a handle:

https://www.amazon.com/Lola-377-Handle-Kitchen-Sponge/dp/B007YPSM54/ref=asc_df_B007YPSM54/?tag=hyprod-20&linkCode=df0&hvadid=167134408017&hvpos=1o7&hvnetw=g&hvrand=14773618084906157864&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9018853&hvtargid=pla-304411950700&psc=1

You can probably find inexpensive ones at Family Dollar or a similar place.

I often spray items like this with scrubbing bubbles bathroom cleaner scrub with a soft brush and rinse well.

I would power wash it and spray it down with some soap