Pregnant women are routinely tested for Down Syndrome by their doctors. It is possible to check for Down in the first trimester of pregnancy using blood tests but they are not very reliable and produce a lot of false positives.
Doctors recommend amniocentesis in the second trimester to screen for Down syndrome and other birth defects. Women over the age of 35 who are more likely to give birth to a child with Down syndrome are highly recommended to have these tests.
It is also possible to diagnose Down Syndrome at birth. A karyotype blood test is ordered if the doctor sees any tell-tale Down signs in the child so that the doctor can determine whether the child has an extra chromosome 21.
Rarely the baby has a mix of normal cells some of which have two chromosomes 21 and some with three.  In children with mosaic Down syndrome or Mosaicism fewer Down symptoms will be present.