Most tourists visit Kenya without any problems at all. However, there has been an escalation in recent years of petty theft, and some aggravated theft on tourists.
Do not carry large sums of money with you, or conspicuous amounts of jewellery/camera equipment etc. Avoid walking at night if possible, especially in larger towns and cities.
However, the British Foreign Office advise against all but essential travel to anywhere within a 60km distance of the Somali border. This corridor also extends down to the coast to just north of Mombasa.
For the coast from Mombasa southward, the island of Lamu (if flying in), and all the safari areas, there is no restriction advised beyond standard security measures.
While on tour in the safari areas, there is little threat of anything more than petty theft. Always keep valuables locked in your hotel safe.
There are sometimes complaints of harassment by “beach boys” of beach goers along the coast. Mostly they are just trying to sell good – just be vigilant, but don’t let is spoil your enjoyment.
As with all other destinations, be aware of your surroundings, and if a situation doesn’t feel right move on.
If you have questions you can follow the official advise of the British Foreign Office here and the US Sate Department here.