Your options are by air or by bus. A taxi from Nairobi to the border, and a new taxi from the border to Arusha is of course also an option, but it will not come cheap.
As mentioned in the answer from @Carl you will have several options for bus companies at around $20. The pick-up for some companies are at the different hotels. The bus will take you to the Arusha bus station. Safety on roads in any country is always a concern. The road between Nairobi and Arusha has been upgraded some years ago. This usually means the average speed goes up. Travelling along roads in African countries is more dangerous than in any European country. Safety in cars and buses are generally lower - lack of safety belts, driver ignorance or lack of skill, bad roads, bad technical standard on cars, high speed driving, securing of cargo, cattle/donkeys along the road etc. A drive will, for better or worse, of course give you a different feel for the countries you travel in and Kenya/Tanzania are no exceptions.
Your cheapest flight option would be with Precision Air. It currently costs around $150. Since it is an international flight it will take you from Nairobi(NBO) to Kilimanjaro international Airport (JRO). The Arusha airport is not an international one. You will have to add the bus ticket from JRO to Arusha. It will set you back from around $10. A taxi will cost you around $50. The distance from JRO to Arusha is around 50 kms, so you will still have to spend an hour or so on the road.