JavaScript (V8), 7369 63 bytes
g=>l=>l.map(w=>Math.min(...[...w].map(c=>g.split(c).length-1)))-4 bytes thanks to emanresu A
-6 bytes thanks to Matthew Jensen
JavaScript (Node.js)+Ramda, 56 bytes
g=>d=>d.map(w=>R.min(...R.map(c=>R.count(x=>x==c,g),w)))JavaScript +Sugar, 52 bytes
g=>d=>d.map(w=>[...w].map(c=>[...g].count(c)).min())Sugar.extend();f=g=>d=>d.map(w=>[...w].map(c=>[...g].count(c)).min())console.log(f("REKNA,TAXIJ,RAREN,ATAEI,YCYAN")(["AIR", "ANKER", "EAT", "CYAN", "NINJA", "RARE", "TAXI", "TRAY", "XRAY", "YEN"]).join(', '));console.log( f("JLIBPNZQOAJD,KBFAMZSBEARO,OAKTMICECTQG,YLLSHOEDAOGU,SLHCOWZBTYAH,MHANDSAOISLA,TOPIFYPYAGJT,EZTBELTEATAZ") (["BALL","BAT","BEAR","BELT","BOY","CAT","COW","DOG","GAL","HAND","HAT","MICE","SHOE","TOP","TOYS","ZAP"]) .join(', '));<script src="https://cdnjs.cloudflare.com/ajax/libs/sugar/2.0.6/sugar.min.js"></script>