Not just demographics, by a long shot. Immigration is a great thing for people and countries throughout the world, provided that immigrants come to a country not just for the social benefits, but to actually add something to the economy. When a country allows immigrants that aren't on the dole, benefit occurrs.
The amount of immigrants should increase to increase the amount of diversity in these countries. A world where people of many different ethnic backgrounds can get together is a better world. This I think will help lessen the amount of Racism in these countries, becoming more excepting of other people.
No, European countries should not attempt to attract more immigrants to "shore up the region's demographics." Many African and Middle Eastern migrants already flock to Europe in droves, with many migrants dying at sea in the Mediterranean or have to be rescued from capsized boats. There are plenty of immigrants with their sights set on Europe already, so for that reason, it is not necessary for Europe to try to attract immigrants.
No, European countries should not attract more immigrants to shore up the region's demographics, because there is already diversity in Europe, and plenty of people. A country does not need to have a large population in order to be wealthy. A small number of people can be prosperous if they have good leadership and a free economy.
European countries, especially if they are in the EU do not have any trouble with a diversification of demographics and immigration. They are free to move around the Euro zone and are able to increase demographics in a lot of countries. There is no addition help that is needed into order to do more.