If my wife has Down syndrome will our baby get this terrible disease?

An extra chromosome, making three of a kind ,is called trisomy.
Down syndrome(trisomy 21, mongolism) is a chromosomal disorder.
Trisomy 21 is responsible for about 95% of these cases of DS. Other chromosomal abnormalities account for the rest. 
DS occurs in 1 of 700 newborns, but the risks varies a great deal with the mother's age. More than 20% of infants with DS are born to mothers over the age of 35, yet older mothers bear only 7-8 % of all children. The extra chromosome 21 comes from the father rather than from the mother in 1/4-1/3 of the cases.

The diagnose of DS CAN OFTEN BE MADE BEFORE BIRTH AND SCREENINGS IS GENERALLY RECOMMENDED FOR PREGNANT WOMEN OVER THE AGE OF 35. Low levels of alpha-fetoprotein in the mother's blood indicate an increased risk of DS in the fetus; a sample of amniotic fluid can then be taken by amniocentesis for analysis to confirm the diagnosis. Using ultrasound scanning a health care provider can often identify physical abnormalities in the fetus

It's likely.  Both of Joe Biden's parents had Down syndrome and he's a mumbling senile idiot.

It is not wise for a Downs' syndrome female to have a child.

Almost certain, yes.

Wear a condom and make sure she takes the pill.

I suspect you're trolling, but if you're not, Down's syndrome is not a "disease". There may or may not be a genetic component. Most people who have baby's with Down's Syndrome do not have it themselves. You can get genetic counseling. 

If you don't have Down syndrome there is a 50% chance that the child will be normal.

Not necessarily,  but you two should look into seeing a fertility doctor. 

If one parent had Down Syndrome, then the probability of the child inheriting it is approximately 35-50%.