It is still used in formal situations, such as between business people in work contexts. Note that it is only used when you meet someone for the first time, and is a more formal equivalent to 'It's nice to meet you' or 'I'm pleased to meet you'. It is NOT an alternative to 'How are you?' which is a question that can be asked on any occasion. Take a look at this post from earlier today: http://www.italki.com/question/321035

I agree that "How do you do," could sometimes be used in formal situations, like in business, but maybe only by people who are middle aged. I can't imagine myself or someone my age using "How do you do" even in the most formal context, we would just say "How are you?" with a more polite tone. My experience is in the U.S. though; I've heard the UK is super formal, so maybe it is more commonly used still over there or in Australia.

Hi Helen! I'm 23 year old too. Uhm! Nice to meet you. My english is very weak then no answer =))