Hello, and welcome to the Forum.
Emerald Green Arborvitae (Thuja occidentalis 'Smaragd') is usually considered to be more a large shrub than a full-sized tree. That it would cause damage to a neighboring basement is doubtful, unless there are truly exceptional circumstances at play here.
Your neighbor may be overly worried, or s/he may really not want the plants there for another reason, such as shade or concern that his or her view will be blocked.
As for substitutes, all other shrubs of similar size will also have root systems of a similar size. Alternatives such as a trellised vine would require more effort for less height, and anyway, larges vines have large root systems too.
Ignoring, educating, or otherwise negotiating with your neighbor seem to be other options to consider.