Question Summary:
Is it permissible for a woman to say salam to non-mahram men?

Question Detail:

Is it permissable for a married women in Pardah to make Salaam in a loud voice when entering a home of non-mahrams/to non-mahrams and elderly male family members? The lady does not look or talk anything else, she does Salaam out of respect loudly and walks off...?

Answer :

In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
Saying salam is indeed a virtuous act and a sunnah of our beloved nabi (sallallahu alaihi wa salam) and the anbiya (alaihimus salam) before him. However, it is the wisdom of Allah the Almighty that he has designed Shari'ah in such a way that it closes all doors leading to fitnah. Allah Ta'ālā says in the Holy Quran:
وَلَا تَقْرَبُوا الزِّنَا إِنَّهُ كَانَ فَاحِشَةً وَسَاءَ سَبِيلًا 
And do not even go near fornication. It is indeed a shame and an evil way to follow. [Al-Quran, 17:32]
Living in an era where male female interaction has become the norm of society, we must tread carefully and try our best to stay away from all situations that might lead to haram. Allah Ta'ālā has blessed women with natural beauty and elegance. Even though a woman might be sincere when extending greetings to the opposite gender, her mere voice has the ability to captivate the hearts of men and attract them towards her as Allah Ta'ālā Himself has mentioned in the Holy Quran:
فَلَا تَخْضَعْنَ بِالْقَوْلِ فَيَطْمَعَ الَّذِي فِي قَلْبِهِ مَرَضٌ وَقُلْنَ قَوْلًا مَعْرُوفًا 
Do not be too soft in your speech, lest someone having a disease in his heart should develop fancies (about you); and do speak with appropriate words. [Al-Quran, 33:32]
Besides the voice factor, in this era of fitna and immorality, the mere greeting of the opposite gender is a means of initiating contact which could lead to informal contact. There have been innumerable instances that bear testimony to the evils emanating through this. As such, it is advisable not to extend salam to non-mahram men[1].
And Allah Ta’āla Knows Best
Bilal Mohammad
Student Darul Iftaa
USA 
Checked and Approved by,
Mufti Ebrahim Desai.
www.daruliftaa.net