The best way is the remove the old grout clean the area and regrout
A clear silicon or a colored grout (in caulking gun tubes) may ne needed to re-seal your joint. Removing existing grout would be best, but a bead at the joint may work.
If it's bad, you may have to remove it and redo the grout but if not, you can likely use some caulk to tidy that up.
Scrape out any loose grout and spread a bead of clear silicone caulk.