There are plenty of nurseries available in Dubai, although demand for places is high and you should try to get your child's name on a nursery's list as soon as practicable. The majority of nurseries do not take children who are under 18 months old. Care is usually provided during 'school hours', i.e. from 8 a.m. until 12.30 p.m.
A comprehensive list of nurseries in Dubai is provided on the Dubai Kidz website.
If both parents are working, it will be difficult to find a nursery that can care for your child full time and you may prefer to consider hiring a maid or nanny. As the employer, you will have to pay around 1000 AED in salary per month and you will be responsible for the cost of obtaining a visa for your employee, who will probably be a foreign worker, which amounts to 5200 AED. It is also normal to pay for an airfare to her home country every two years. The visa must be renewed (and the government fee paid again) each year.
A number of agencies offer part time home help on an hourly basis, which may prove to be a more economical option. The charges are usually in the region of 20 AED per hour.
If you are looking for childcare for the evening, the simplest and safest solution is to ask your neighbours and friends. The classified advertisements sections of local newspapers may have advertisements from students seeking babysitting work to earn a little extra money.