In his letter Mr Johnson outlines the "deep concerns" the BMA has with recent problems which have "caused anger and anxiety throughout the medical profession, and [have] diminished doctors' confidence in the reform of medical training".
Both the Medical Training Application Service (MTAS) and Modernising Medical Careers (MMC) systems have been plagued with difficulties.
This week the MTAS website was shut down by the Department of Health after it emerged that personal details of applicants could be accessed, including telephone numbers, addresses and sexual orientation.
"Such widely publicised technological failings in a relatively simple IT system must… have the potential to damage patients' confidence in the security of the care record that the Department for Health is introducing," Mr Johnson said.
And earlier this month a survey by the BMA found that 50 per cent of junior doctors are likely to move abroad for work if they cannot get a job through MTAS.
Problems with the MMC have forced a review of the plans and led to health secretary Patricia Hewitt apologising to doctors.
Motions at the BMA's junior doctors' conference today include a vote of no confidence in the government team implementing changes to the recruitment process.
"The BMA has raised… issues consistently and directly with the responsible ministers and officials and has raised serious concerns about many aspects of the system for more than a year and called for a delay to the implementation last July," Mr Johnson's letter claims.
"I hope that you will personally bring your influence to bear on achieving an urgent and comprehensive solution to these serious problems.
"Doctors are rightly angry and frustrated and will grow increasingly so if these issues are not addressed with the level of urgency they deserve," he concludes.
Click here to run an energy price comparison, and see if you could be paying less for your gas and electricity.