Re the blue eyes - even if the baby ends up with blue eyes (but they may well change colour, my 3 brothers and sisters and I all started with blue, we now have green (2 of us), grey and blue respectively), that is fine...
Basically, blue (and green) eyes are recessive genetically.
That means that if a person has genes for both blue/green and brown eye colour, their eyes will be brown because the brown is dominant over the blue/green. They will however be carrying a blue/green gene and can pass this on to children.
[Some people with brown eyes will of course be carrying two brown genes, so will have no blue/green to pass on.]
Because blue/green is recessive, for someone to actually have blue or green eyes, they must have two blue/green genes and no brown, as the brown would dominate.
As a result, two brown eyed parents can have a blue eyed child, because they may both be carrying and have passed on the blue/green gene. (Hence ant3nna's mum!)
However, two blue/green eyed parents cannot have a brown eyed child - the parents have no brown gene to pass on.
You have brown eyes and your partner has green, right? Thus you could have a brown eyed child (because you would have passed on your brown gene, which is dominant) or a blue/green eyed child (becasue you are carrying a blue/green gene) and you partner can only pass on blue/green (becasue he carries no brown).
If it is the fact that the baby has blue eyes rather than green that you are worried about, really don't worry as babies' eyes often change colour. My youngest brother had blue eyes up to the age of about 3 and now (aged 15) his eyes are properly green, like mine.