Thanks for your question about the health of your alder tree. Without seeing the tree, it is impossible for me to say if it is in danger of falling. If it looks healthy, that is a good sign. It is normal far small, and even fairly large, branches to die and drop, especially in alders. Healthy trees rarely just up and fall over, but a bad storm or combination of weather events can knock even a healthy tree over.
The most important question is, if it fell, what would it fall on? There is a big difference in the risk level of a tree fall, between it falling on your house or other place people frequent, and falling in, say, an open lawn.
The only way to tell for sure how healthy your tree is, and evaluate the risks of it falling, is to consult a licensed arborist. They can evaluate its health, suggest any care it might need, and estimate where it would be likely to fall if it did fall.
It is always a shame to cut down a mature tree, which provides rare habitat for many small creatures in urban areas, as well as cooling shade in our warming climate. Sometimes it is necessary because the tree is diseased, damaged, or otherwise causing a risk. But, sometimes there are other reasons people want to cut down a tree - it casts too much shade, it is messy, or they are just tired of raking leaves. Could one of these other things be the real reason your wife would like to see it gone? If so, perhaps you two can discuss the benefits vs the problems of the tree, and find a way to minimize the problems (i.e., hire someone to rake leaves, or have an arborist thin it to increase light, and so on).
Good luck, and I hope you can keep your tree!