The short answer is no. Children are incredibly sensitive to the different ways people speak. Even when they only hear one language, they learn very quickly about differences between the way men and women talk, the difference between polite and impolite ways of talking, and so on. For children, the bilingual situation is just a matter of another difference between people!
Fifty years ago educators throughout North America used to tell immigrant parents that it was better for their children's schooling if they spoke English at home. Some researchers thought that early exposure to two languages put children at a disadvantage. Newer research tells us that this is not so, and there may be advantages to being bilingual (in addition to knowing more than one language), such as more flexible thinking. The disadvantages that earlier research found were generally economic disadvantages, linked to the hardships of immigrants' lives.
Bilingual development sometimes results in slightly slower language development than for some monolingual children. Our older child was still saying things like Where you are? instead of Where are you? in English at four and a half. This is a normal developmental stage for monolingual English children, but they usually figure out that they have to say Where are you? by the time they're three or four. Our older child just took a little longer.
-Linguistic Society of America