Technically, you can prepare your BAS yourself, going through all your invoices and receipts and tallying the GST that you have received/paid, and the total income that you have received (this is the Simpler BAS). If you are a small business, you generally do this on a cash basis (ie when the money has been banked/spent).

An accountant or bookkeeper is advisable. A BAS agent is someone registered with the Tax Practitioners Board to be able to complete BAS and charge for their services, a Tax agent does what a BAS agent does, but also tax returns. It is very important that whomever you are using is registered, and you should check with the Tax Practitioners Board that they are. You can search the Board here:
https://www.tpb.gov.au/registrations_search

In terms of cost, that is up to the accountant or bookkeeper, and generally depends on how much work they have to do.

Hi, You can complete your own BAS, to need to check that all your bank transactions are posted and that the balances agree and the correct tax codes have been used when recoring the transactions in your accounting system.

In regards to  BAS agents these are professional people who focus on Business Activity Statement work, GST etc and the core bookkeeping functions ( The programs like XERO, MYOB & QBO etc), where an accountant is mainly focused on Income, Company  Tax and also GST , you need to make sure that they are regsitered withe TPB. The cost to process a Business Activity Statement will vary, in relation to amount of corrections to entries, the amount of transactions to check, whether the bank accounts reconciles. You can be assured that by using a TPB registered BAS or TAX agent will have specific checklists to work through, to ensure the return is up the required standard.
 

Hi,
Only registered BAS and TAX AGENTS with TPB can prepare and lodge your BAS.
You must  check their registerations before proceeding.
The fee charged for your BAS return is calculated on the time required to prepare it and what kind of accounting system is being used by  the tax payer.
 

A bookkeeper needs to be a BAS agent to provide BAS services. You can easily check if they are registered, as Kristy mentioned. This is only if you are outsourcing your bookkeeping to a third party. If you are preparing and lodging BAS statements yourself, you don’t need to be a BAS agent.
 
In short, you can use an accountant or a bookkeeper who is a BAS agent. Accountants may not do it themselves. They and accounting firms typically have bookkeepers who will work on the BAS statements and lodge it for them. The cost should be around a few hundred dollars for a small business.