Two common objections to devotion to the Blessed Virgin, in my experience, are "other people make Mary into a demi-God, so I won't have any devotion to her" which is the common fallacy of "abusus non tollit usus" aka "throwing the baby out with the bathwater" and "devotion to Mary competes with devotion to Christ". I have even heard people go so far as to say "Not only does devotion to Mary hinder a relationship with Christ, but it is pagan idolatry." Let us answer each of these objections in turn.
I know that I personally fell into the first fallacy when it came to the pro-life movement several years back. I was pro-life, but didn't want to get involved because I thought of all those people who just yell at women at clinics as the pro-life movement, and I didn't want to be associated with that. I saw those people as doing more harm than good. However, I realized, with the aid of some friends, that there were plenty of good things within the pro-life movement, such as crisis pregnancy centers, respectful protest, sidewalk counseling, educational materials, etc. I also realized that I was cheating myself by letting a few bad apples deprive me of involvement in something good. I think it is the same way with those who fall into the first fallacy when it comes to devotion to the Blessed Mother. Just because a few bad apples take it too far does not mean we should deprive ourselves of involvement in something good. In fact, it means that we should be a "light" by practicing proper devotion to our Blessed Mother.
The second objection is more common, but, in my opinion, just as illogical. First of all, in MY experience, the people I know who have a visible Marian devotion (ie wear a scapular, pray the Rosary, etc.) also attend daily Mass and Eucharistic Adoration at least as much – if not more – than their non-Marian counterparts. That could just be my experience, though. Let us take a look at history. If we are called to be more Christ-like, we should strive to imitate Him in all things. He loved His Mother and "became subject to [Mary and St. Joseph]" (Lk 3:51) If she's good enough for Him, shouldn't she be good enough for us? The marriage feast at Cana (Jn 2:1-10), Jesus' first public miracle, reflects both how devotion to Mary is Christ-like and how devotion to Mary does not compete with devotion to Christ but leads us closer to Christ. When Mary points out the lack of wine, Jesus – although He says that "His hour has not yet come" (Jn 2:4) – answers Mary's request by making wine that was better than what the guests had already consumed (Jn 2:10). Mary also provides the perfect instructions for us by telling the stewards "'Do whatever He tells you'" (Jn 2:5). She leads us closer to Christ by her example. She never takes any credit for herself; she submits to God's will and always directs the glory to Him. At the Annunciation, she gave us the perfect model of response to God, "Behold the handmaid of the Lord; be it done to me according to thy word" (Lk 1:38). At the visitation, when Elizabeth gave Mary honor,
"'Blessed art thou among women and blessed is the Fruit of thy womb! And how have I deserved that the mother of my Lord should come to me? For behold, the moment the sound of thy greeting came to my ears, the babe in my womb leapt for joy. And blessed is she who believed, because the things promised her by the Lord shall be accomplished'" (Lk 2:42-45)
Mary's response was one of humility, giving all the glory to God.
"And Mary said, 'My soul doth magnify the Lord. And my spirit hath rejoiced in God my Saviour. Because He hath regarded the humility of His handmaid; for behold from henceforth all generations shall call me blessed. Because He that is mighty, hath done great things to me; and holy is His name. And His mercy is from generation unto generations, to them that fear Him. He hath shewed might in His arm: He hath scattered the proud in the conceit of their heart. He hath put down the mighty from their seat, and hath exalted the humble. He hath filled the hungry with good things; and the rich He hath sent empty away. He hath received Israel his servant, being mindful of His mercy: As He spoke to our fathers, to Abraham and to his seed for ever.'" (Lk 1:46-55)
Mary leads us to a greater devotion to her Son, keeping none of the glory for herself.