Several recent studies suggest that a child born into a family whose

(A) a child born into a family whose members have allergies will probably themselves develop allergies
=> mismatch as "a child" is a singular noun while "themselves" is a plural reflexive pronoun

(B) children born into families whose members have allergies will probably themselves develop allergies
=> correct use of relative pronoun "whose" and reflexive pronoun "themselves" which refers to "children"

(C) a child born into a family the members of which have allergies will probably develop an allergy
=> the use of "of which" is not approriate => a "whose" between "family" and "members" is preferred

(D) in those families where members have allergies, children will probably develop allergies themselves
=> "those" is quite a redundant & the relative pronoun "where" is not approriate here as "where" refers to place

(E) children born into a family in which there are allergies will themselves probably develop an allergy
=> the use of "in which" is not correct as "in which" refers to place & "there are allergies" sounds awkward as if the allergies are not of family members.