Should one use the definite article before a job title followed by a name?

I think it would have been better if they had printed either of the following.

• Warm greetings to the President of India, Pranab Mukherjee
• Warm greetings to the President of the Republic of India, Pranab Mukherjee

The first one sounds good because there is no repetition of "the". The second one is correct, but it doesn't sound so good as the first one.

You need to use "the" to refer to unique things (though people tend to omit it often).

Examples

• The sun rose at 6:17 this morning.
• You can go anywhere in the world.
• Clouds drifted across the sky.
• The president will be speaking on TV tonight.
• The CEO of Total is coming to our meeting.

— EF.com

The basic question is whether you are saying "Greetings to [honorific][human's name]" or "Greetings to [official title], [human's name]".

You would say "Greetings to the President of the Republic of India", but "Greetings to Pranab Mukherjee".

Combine them one way and you get "Greetings to the President of the Republic of India, Pranab Mukherjee". Combine them the other way and you get "Greetings to President of the Republic of India Pranab Mukherjee", where "President of the Republic of India" is an honorific (just like "Mr." or "General" or whatever).

The banner is correct as it appears, so long as it contains no comma. It might read a hair better the other way (with comma), but protocol may dictate against that.