Washing your hands at regular intervals could be the most effective method to staying healthy and protecting yourself from various ailments.

(I) is the most effective way of staying

(II) can be the most effective way to stay

(III) has the effective effect to stay

“is the most effective way of staying” is the correct phrase to make the sentence grammatically correct as the sentence is in generalized and factual form. Thus the use of “could” or “can” is incorrect in this case. Moreover, the phrases connected with the conjunction “and” should be in similar form. Thus among the given options, only option (I) has the correct grammar structure to correctly fit into the sentence. Hence 1 is the correct option.