• If you think you will out in the open for too long and may not have ready access to a bathroom, a handkerchief (for formal situations) or a small hand towel (if you are going to hike or workout or be outdoors) are a must.  That way you can use it to wipe yourself if you sweat or if you end up washing yourself in the open.
• While many people may know this but let me repeat - no matter in what culture you are, it is simply not a good idea to blow your nose into a napkin.  What I suggest is that if you need to blow your nose and rush to the nearest bathroom.  You should always blow your nose in your own handkerchief and be as discreet about it as possible.  You should do the same after sneezing.
• In many poor countries where public restrooms do not have paper towels or tissue paper, you will find that having a handkerchief is very helpful.  Similarly, if you are traveling in an area where you likely to experience a lot of dust, you can use a handkerchief to cover your nose so that you do not breathe a lot of dust particles.

Handkerchief etiquette

• Use it only once.  As soon as you can, please change it.  If possible, carry more than one, particularly if you will be gone for several hours or if you have a cold.
• While silk handkerchiefs are great for style, it is best to have a cotton handkerchief for actually using it.
• There is no rule that men should carry only dull, white handkerchiefs but it is generally expected that men will not be using the types popular among women (frilly, flower-patterns, feminine colors, etc.)

Related article:  How to be classy?  Manners and etiquettes