I've noticed the therapeutic effect of herbs seems to vary hugely depending on the brand. I guess because the FDA does not regulate the production, we get really significant differences in quality. Allergy Research is generally a good brand, but I can't comment on their Artemesinan since I've not taken that particular product.
Generally though any abrupt increase in your Babesiosis symptoms would suggest it's working for you. I've taken Art from few different manufacturers and didnt' observe any sie effects, just the Herx symptoms. I'd start with at least 300mg daily in 100mg divided doses, but you might need twice that to see any benefit. After about 3 week take a week off, then resume. Art tends to not be effective in small doses, and there's some enzyme that builds up that prevents it's absorption about about 3 weeks of dosing and the one week break avoids that problem.