According to a survey conducted in 2008, there are about 4 babies born every second. That's 240 every minute and so you would have to measure time in units finer than seconds if everyone were to have a unique prime number associated with themselves using this method. I've determined that 65552341949 (corresponding to 6:55:52am on the 3rd April 1949) is prime but that's only going down to the second level. Of course, in my day the birth rate was much lower because there were only 2.5 billion people in the world, about a third of the current total, but there's still not enough to go around. Hundredths of seconds is the only way to do it.
There are other ways to represent dates of course. I've used DD-MM-YYYY but MM-DD-YYYY and YYYY-MM-DD are also possible. It seems to me that it's best to put the year first and then work down from there. That way, it can be seen at a glance what year a person was born. On this basis, I could lay claim to 1949040306570101 corresponding to 6:57:01:01am on the 3rd April 1949. That's good enough for me and so I'm laying claim to it.
ADDENDUM: added January 11th 2019
After re-reading this post from June 26th 2013, I'd propose these modifications while retaining the YYYYMMDDHHMM sequence (with a twenty four clock used for time). If the time of birth is not known then midday 12:00 should be used. The remaining four digits are to chosen so that the resulting number is prime, beginning with 0001 if it is not taken and proceeding up to 9999. There is a little under 300 primes in this range of 9999 numbers. Currently, the birth rate is about 4.3 per second or 258 per minute (source) and declining so the four digits are sufficient. In this scheme the first possible number for me would be 1949040306570059 which by the way is a palindromic prime (9500756030409491 is also prime). There are 282 suitable candidates. Below is the full list:
1949040306570059
1949040306570061
1949040306570067
1949040306570083
1949040306570101
1949040306570163
1949040306570167
1949040306570179
1949040306570229
1949040306570251
1949040306570277
1949040306570289
1949040306570361
1949040306570367
1949040306570377
1949040306570491
1949040306570493
1949040306570517
1949040306570527
1949040306570589
1949040306570619
1949040306570653
1949040306570769
1949040306570779
1949040306570817
1949040306570839
1949040306570893
1949040306570923
1949040306570937
1949040306570941
1949040306570961
1949040306571079
1949040306571109
1949040306571133
1949040306571291
1949040306571301
1949040306571331
1949040306571363
1949040306571409
1949040306571439
1949040306571447
1949040306571481
1949040306571487
1949040306571541
1949040306571571
1949040306571669
1949040306571691
1949040306571693
1949040306571697
1949040306571789
1949040306571817
1949040306571823
1949040306571843
1949040306571847
1949040306571873
1949040306571907
1949040306571997
1949040306572053
1949040306572063
1949040306572083
1949040306572129
1949040306572131
1949040306572141
1949040306572147
1949040306572173
1949040306572179
1949040306572287
1949040306572441
1949040306572479
1949040306572501
1949040306572503
1949040306572509
1949040306572533
1949040306572587
1949040306572609
1949040306572651
1949040306572663
1949040306572743
1949040306572773
1949040306572813
1949040306572857
1949040306572861
1949040306572911
1949040306572917
1949040306572941
1949040306572951
1949040306572969
1949040306573029
1949040306573131
1949040306573137
1949040306573149
1949040306573221
1949040306573263
1949040306573307
1949040306573337
1949040306573343
1949040306573377
1949040306573433
1949040306573457
1949040306573461
1949040306573467
1949040306573553
1949040306573611
1949040306573641
1949040306573653
1949040306573677
1949040306573713
1949040306573739
1949040306573749
1949040306573781
1949040306573793
1949040306573821
1949040306573881
1949040306573901
1949040306573959
1949040306574001
1949040306574003
1949040306574057
1949040306574073
1949040306574079
1949040306574087
1949040306574099
1949040306574129
1949040306574133
1949040306574241
1949040306574301
1949040306574327
1949040306574391
1949040306574411
1949040306574433
1949040306574507
1949040306574511
1949040306574523
1949040306574567
1949040306574579
1949040306574597
1949040306574639
1949040306574793
1949040306574849
1949040306574859
1949040306574883
1949040306574939
1949040306574969
1949040306574973
1949040306574981
1949040306575107
1949040306575111
1949040306575113
1949040306575143
1949040306575213
1949040306575279
1949040306575309
1949040306575347
1949040306575407
1949040306575477
1949040306575513
1949040306575543
1949040306575591
1949040306575623
1949040306575701
1949040306575723
1949040306575729
1949040306575743
1949040306575753
1949040306575779
1949040306575797
1949040306575807
1949040306575839
1949040306575849
1949040306575903
1949040306575963
1949040306576001
1949040306576037
1949040306576043
1949040306576113
1949040306576161
1949040306576179
1949040306576289
1949040306576347
1949040306576367
1949040306576371
1949040306576443
1949040306576469
1949040306576491
1949040306576527
1949040306576599
1949040306576613
1949040306576617
1949040306576619
1949040306576673
1949040306576689
1949040306576707
1949040306576709
1949040306576719
1949040306576799
1949040306576817
1949040306576859
1949040306576899
1949040306576991
1949040306577003
1949040306577037
1949040306577061
1949040306577073
1949040306577079
1949040306577093
1949040306577207
1949040306577291
1949040306577349
1949040306577361
1949040306577393
1949040306577499
1949040306577513
1949040306577547
1949040306577589
1949040306577607
1949040306577631
1949040306577691
1949040306577729
1949040306577787
1949040306577831
1949040306577877
1949040306577927
1949040306577981
1949040306578011
1949040306578059
1949040306578063
1949040306578179
1949040306578243
1949040306578251
1949040306578269
1949040306578293
1949040306578341
1949040306578347
1949040306578389
1949040306578419
1949040306578423
1949040306578527
1949040306578543
1949040306578587
1949040306578593
1949040306578609
1949040306578641
1949040306578819
1949040306578839
1949040306578843
1949040306578917
1949040306578933
1949040306579007
1949040306579083
1949040306579161
1949040306579169
1949040306579229
1949040306579241
1949040306579257
1949040306579259
1949040306579263
1949040306579283
1949040306579287
1949040306579293
1949040306579307
1949040306579341
1949040306579389
1949040306579439
1949040306579449
1949040306579511
1949040306579521
1949040306579529
1949040306579547
1949040306579557
1949040306579559
1949040306579563
1949040306579577
1949040306579581
1949040306579629
1949040306579643
1949040306579661
1949040306579809
1949040306579829
1949040306579833
1949040306579887
1949040306579913
1949040306579943
total is 282